UNIVERSITY  OF  CALIFORNIA 
AT    LOS  ANGELES 


SCLLIVflN'S 

NEW    HYDRAULICS 

Consisting  of 

New  Hydraulic  Formulas 

and 

The   Rational   Law  of  Variation  of 
Coefficients 


Plow  and  Resistance  to  Flow  in  all  Classes  of  Rivers,  Canals, 
Flumes,  Aqueducts,  Sewers,  Pipes,  Fire  Hose.  Hy- 
draulic  Giants,   Power   Mains,   Nozzles, 
Reducers,  etc.   with  Extensive    Ta- 
bles and  Data  of  Cost  of  Pipes 
and    Trenching     and    Pipe 
Line  Construction. 


MARVIN  E.  SULLIVAN,  B.  PH.,  L.L.B, 
Hydraulic  Engineer. 


DENVER 

MINING  REPORTER 
1900 


COPYRIGHT,    19OO 

BY 
MARVIN     E.     SULLIVAN 


C. 


SULLIVAN'S  NEW  HYDRAULICS. 


Page  36— Under  table  of  channels,  add  the  following: 
Divisor  :  quotient  ::  Dividend  :  quotient  squared. 

P        :        r         ::        a          :        r8 
or  r        :        r8         ::        p         :        a,  in  any  possible  case, 


Page  40— 9th  line  from  bottom.  "If  the  value  of  the  co- 
efficient etc."  This  should  be  corrected  so  as  to  read: 
As  the  loss  of  effective  head  or  slope  is  inversely  as  v/d3 
or  i/r3  for  any  constant  head,it  is  evident  that  a  change  in 
the  value  of  d  or  r  cannot  affect  the  value  of  the  coeffi- 
cient n,  for  as  the  loss  of  head  per  foot  length,  S",  de- 
creases directly  as  -\/d3  or  -y/r3  increases,  the  effective 
head  or  slope  will  increase  as  /d3  or  ^/r3  and  this  will  re- 
sult in  a  like  increase  of  v2.  As  n  is  the  ratio  of 
SVd8 


,  and  as  S"  varies  inversely   with   v/ds,  it  is  evi- 


dent that  vs  varies  directly  with   v/ds    or  yr3.      Hence 

the  ratio  of  — 5- — —  n,  will  be  constant  for  all  diameters 
v2 

and  all  slopeu  and  velocities,  and  will  not  be  affected  by 
anything  except  a  change  in  roughness  of  perimeter.  A 
similar  correction  should  be  applied  to  similar  errors  oc- 
curring from  page  40  to  page  51.  See  for  a  general  cor- 
rection of  such  errors,  pages  237  to  241. 


42—     /_ L=  /    ^      should  be  C=    '-L 
V    m        vS-i/r3  A  m 

°'C=4^T_ 

Page  42— v=Cf/r3  y<S  should  be  v=C  t/r3  Xi/S 
Page42— 9th  line  from    bottom,  "Varies   with  {/r3"  should  be 


7,, 


SULLIVAN'S  NEW  HYDRAULICS. 
"Varies  inversely  with  $/r8.    But  see  pages  237  241. 

Page  45 — 18th  line  from  top.  "His  n  might  be  made  &  o" 
should  be  "His  C  might  be  made  to  vary  &c. 

Page  46— Bottom  line,     v3  should  be  v8. 

Page  48— 9th  line  from  top.    S/r2  should  be  S/r8. 

Page  52— Equation  (18)  should  be  m=  s'+Sv  Xy/r8 

Page  85— River  Elbe.  C=49.80  should  be  0=32.51,  and  C= 
52  should  be  0=38.40. 

Page  86— Top  line.  1885  should  be  1855. 

Page  109,  110— Remark.  Add  the  following:— As  large  cast 
iron  pipes  are  full  of  swellings  and  contractions, a  48  inch 
pipe  is  not  really  48  inches  diameter.  As  the  effective 
value  of  any  constant  head  or  slope  varies  directly  with 
the  actual  value  of  ;/d3,  if  we  credit  the  pipe  as  being  48 
inches,  when  in  fact  it  is  not,  the  result  is  to  make  the 
value  of  Sv/d3,  apparently  too  large  for  the  correspond- 
ing value  of  va.  Hence  m  will  be  too  large  or  C  will  be 
too  small  for  a  pipe  which  is  really  48  inches  diameter 
throughout.  The  irregularity  of  diameter  in  large  cast 
iron  pipes  reduces  the  value  of  C  by  from  3  to  5  per  cenf 
for  diameters  greater  than  36  inches.  It  is  not  the  fault 
of  the  formula,  nor  is  it  a  peculiarity  in  the  law  of  flow. 
It  is  simply  the  fault  in  casting  pipes  of  large  diameters. 


Page   114—  4th    line    from    bottom.     ACXt/d3  -3-    should 


be  A  CXt/d3=  -5L 


Page  148-d=U/«  gMgOBXq*  2632^^-  shouM  be 


PREFACE 

|_|YDRAULICIANS  and  engineers  have  long  been  aware 
•  *  that  there  is  some  element  or  law  governing  the  flow 
and  resistance  to  flow  of  water  which  is  not  provided  for  in 
any  of  the  formulas  presented  up  to  this  time.  This  is  made 
evident  by  the  fact  that  the  results  as  computed  by  formula 
do  not  agree  with  actual  results,  and  by  the  further  fact  that 
no  two  formulas  will  give  the  same  result  for  like  conditions. 
Nearly  all  writers  give  one  theory  of  flow  and  resistance  to 
flow  in  pipes  and  closed  channels,  and  an  entirely  different 
theory  and  formula  for  flow  in  open  channels.  The  usual 
formulas  for  flow  in  pipes  give  results  too  high  for  all  diam- 
eters smaller  than  about  fourteen  inches,  and  too  low  for 
all  greater  diameters. 

The  loss  of  head  or  pressure  by  resistance  as  computed 
by  the  ordinary  formulas  is  much  too  small  in  small  pipes  and 
greatly  in  excess  of  the  truth  in  large  pipes.  The  reason  of 
these  erroneous  results  is  that  the  coefficients  were  deter- 
mined for  pipes  of  medium  diameter  and  do  not  vary  correctly 
so  as  to  meet  the  requirements  of  varying  diameters.  Hence 
the  greater  the  diameter  varies  from  the  medium,  either  be- 
low or  above,  the  greater  will  be  the  error  in  the  computed 
result.  The  usual  formulas  for  flow  in  open  channels  are 
equally  faulty,  but  not  in  the  same  way. 

The  acknowledged  fault  in  all  the  formulas  so  far  pre- 
sented is  due  to  the  failure  of  hydraulicians  to  discover  the 
rational  law  of  variation  of  the  coefficient.  This  law  has 
been  sought  in  vain  since  the  beginning  of  the  sixteenth  cen- 
tury, at  which  time  Galileo  discovered  the  law  of  gravity  and 
undertook  to  formulate  rules  for  flow  in  rivers.  The  failure 
to  discover  the  true  law  of  variation  of  the  coefficient  has 
been  generally  conceded  by  all,  and  the  possibility  of  its  dis- 
covery has  been  denied  by  macy.  Ganguillet  and  Kutter  ob- 


4  SULLIVAN'S  NEW  HYDRAULICS. 

serve  that  "more  than  a  century  ago,  Michelotti  and  Bossut 
established  the  true  principle  that  the  formulae  for  the  move- 
ment of  water  must  be  ascertained  from  the  results  of  observa- 
tion, and  not  by  abstract  reasoning."  In  the  Translator's 
preface  (Hering&  Trautwine)  to  the  work  of  Ganguillet  and 
Kutter  the  following  observation  occurs: 

"As  V=C  i/RS  will  most  likely  remain  the  fundamental 
expression  for  such  formulae,  the  attention  of  hydraulicians 
will  be  turned  chiefly  to  the  more  accurate  determination  of 
the  variable  coefficient  C.  A  number  of  authors  have  en- 
deavored to  establish  laws  for  its  variation,  and  among  them 
Ganguillet  and  Kutter  appear  so  far  to  have  been  the  most 
successful."  Ganguillet  and  Kutter,  however,  do  not  claim  to 
have  discovered  the  true  law  of  variation,  as  many  unthink- 
ing persons  have  supposed,  but  on  the  contrary  they  announce 
the  belief  that  it  cannot  be  discovered  by  abstract  reasoning, 
and  in  default  of  its  discovery  they  propose  a  formula  which 
is  entirely  empirical.  They  observe  (page  105):  "The  form- 
ula (Kutter's)  rests  only  upon  actual  guagings.  *  *  *  Be- 
an empirical  formula,  it  is  confined  to  the  limits  occurring  in 
nature  and  makes  no  claim  whatever  to  absolute  perfection. 
In  spite  of  the  large  number  of  available  guagings,  it  cannot 
be  denied  that  our  knowledge  of  the  elements  and  laws  of  the 
motion  of  water  still  need  extension  and  correction." 

Webster's  definition  of  the  word  empirical  is,  "used  and 
applied  without  science."  The  Ganguillet  and  Kutter  formula 
does  not  claim  to  be  scientific  or  rational,  and  yet  it  is  a  con- 
siderable improvement  on  some  of  the  older  formulas,  but  is 
quite  complicated  and  not  simple  and  easy  of  application.  In 
the  search  for  the  true  law  of  variation  of  the  coefficient 
the  greatest  mistake  has  been  made  in  assuming  that  V= 
C^/RS  is  the  fundamental  expression  for  the  formula.  The 
I/  S  is  the  factor  which  expresses  the  unimpeded  and  con- 
stant effect  of  gravity,  while  R  is  an  expression  for  a  factor 
which  modifies  the  effect  of  gravity.  The  constant  effect  of 
gravity  should  be  expressed  separately  and  should  not  be  con- 
fused in  the  same  expression  with  other  factors  which  are 


SULLIVAN'S  NEW  HYDRAULICS.  6 

variable  and  which  impede  or  modify  the  effect  of  gravity. 
All  the  variable  factors  should  be  included  in  the  coefficient 
formula  for  C,  and  then  all  elements  which"  affect  or  modify 
the  law  of  gravity  will  be  included  in  the  value  of  C,  and  the 
fundamental  form  of  the  formula  then  becomes  simply  V= 

(V~s. 

This  is  evidently  true,  because  the  value  of  B  or  D  ha& 
no  connection  with  the  law  of  gravity,  which  is  constant. 
The  value  of  R  or  D  has  to  do  directly  with  the  law  of  resist- 
ance, and  as  R  or  D  varies  the  resistance  to  flow  varies.  C  is 
supposed  to  include  this  resistance,  and  hence  the  value  of  R 
or  D  should  be  included  in  the  formula  for  C.  If  there  were 
no  resistance_to  flow  whatever,  then  the  velocity  would  be  di- 
rectly as  i/ "S~  regardless  of  the  value  of  R  or  D.But  as  the  re- 
sistance to  flow  does  vary  with  the  value  of  R  or  D,  it  is  evi- 
dent that  C  must  vary  with  R  or  D,  and  if  we  write V=Cy  ~S~ 
simply,  we  thereby  have  all  modifying  factors  included  in 
the  value  of  C,  while  the  constant  law  of  gravity  is  expressed 
by  i/S.  Thus  we  prevent  confusing  the  opposing  laws  of 
gravity  and  of  resistance  in  one  combined  factor,  and  clear 
the  way  for  ascertaining  the  true  law  which  governs  the  var- 
iation of  the  value  of  C. 

By  the  law  of  gravity  we  know  that  if  there  were  no  re- 
sistance whatever  the  velocity  would  be  equal  in  all  diame- 
ters, regardless  of  dimensions,  where  the  values  of  ^/  S 
were  equal.  But  by  experiment  we  tind  that  the  smaller  the 
diameter  becomes,  the  smaller  the  velocity  becomes  for  equal 
values  of  |/  S  .  It  is  therefore  evident  that  the  resistance 
to  flow  must  vary  with  some  function  of  the  diameter  or  of 
the  hydraulic  mean  radius.  As  the  resistance  to  flow  is  the 
variable  factor,  and  is  separate  and  distinct  from  and  direct- 
ly opposed  to,  the  acceleration  of  gravity  which  is  a  con- 
stant, we  know  that  -\/  S  has  nothing  to  do  with  the  value 
or  variation  of  C.  We  therefore  narrow  the  field  of  investiga- 
tion by  writing  V=C^  S,  and  then  searching  for  the  law  of 
variation  in  C,  which  we  know  is  some  function  of  the  diam- 
eter or  hydraulic  radius.  After  years  of  diligent  experiment 


6  SULLIVAN'S  NEW  HYDRAULICS. 

and  observation  the  writer  discovered  that  for  any  constant 
degree  of  roughness  of  wetted  perimeter,  either  in  pipes  or 
open  channels,  the  value  of  C  varied  asi/R3  or  as  f/D3 
In  other  words,  if  K  is  a  constant  which  represents  the  given 
degree_p£  roughness  of  perimeter,  then  C=KX  t/R3,and  V 
=  GI/S.  But  if  we  confuse  the  element  of  resistance,D  or  R 
with  the  element  of  acceleration  |/~S7by  writing  V=Cv/RS 
then  we  find  C  varies  as  f/R~ simply,  and  if  K  represents 
the  degree  of  roughness,  and  we  write  V=Ci/BS,  then  we 
have  C=KXf/T^andV=<VRS. 

As  all  hydraulic  formulas  are  necessarily  based  on  the 
laws  of  gravity  and  of  friction,  the  correctness  of  such  formu- 
las must  depend  upon  their  accordance  with  these  laws.  If 
our  present  understanding  and  acceptance  of  these  laws  is 
correct,  then  any  formula  which  violates  either  of  these  laws 
must  necessarily  be  incorrect.  It  is  the  object  of  thie  vol- 
ume to  present  the  rational  law  of  variation  of  the  coefficient 
in  accordance  with  our  present  understanding  of  the  laws  of 
gravity  and  friction.  If  those  laws  are  yet  unknown  it  must 
remain  for  some  future  investigator  to  supply  a  theoretically 
correct  hydraulic  formula.  The  discussion  in  the  following 
pages  is  based  on  the  assumption  that  those  laws  are  cor- 
rectly known,  or  nearly  so. 

In  the  case  of  open  channels  and  rivers  of  irregular  cross 
section,  and  where  the  banks  alternately  diverge  and  con- 
verge, and  the  perimeter  varies  in  roughness  at  different 
depths  of  flow,  the  correct  application  of  any  formula  will  be 
difficult.  In  sections  13  and  83  methods  are  pointed  out  for 
ascertaining  the  value  of  C  in  such  cases. 

The  law  of  resistance  in  nozzles  and  convergent  pipes,  as 
herein  stated  has  been  very  thoroughly  tested  and  its  cor- 
rectness established  by  hundreds  of  experiments.  This  law 
will  be  found  cf  great  service  to  hydraulic  miners  and  fire- 
men and  also  in  determining  the  coefficient  for  flow  in  con- 
verging reaches  of  rivers  and  other  channels. 

In  the  course  of  experiments  of  the  writer  which  has 
extended  over  a  period  of  six  years,  it  was  discovered  that  the 


SULLIVAN'S  NEW  HYDRAULICS.  7 

discharges  over  weirs  and  through  orifices,  as  computed  by 
the  usual  formulas,  and  with  the  tabulated  direct  coefficient 
were  frequently  erroneous,  especially  if  the  weir  used  did  not 
correspond  exactly  in  breadth  and  depth  with  that  from 
which  the  coefficient  was  determined.  Interpolation  for  in- 
termediate conditions  was  certain  to  result  in  error.  On  in- 
vestigation it  was  found  that  the  difficulty  lay  in  the  fact  that 
the  law  of  variation  of  the  c;efficient  of  contraction  has  never 
been  discovered.  In  order  to  avoid  this  difficulty  until  the 
law  of  contraction  shall  be  understood,  an  appendix  has 
been  added  to  this  volume  in  which  the  difficulties  are 
pointed  out  and  the  suggestion  made  that  the  position  of  the 
weir  be  reversed  in  order  to  prevent  contraction  from  taking 
place  at  all.  New  weir  and  orifice  formulas  are  proposed  and 
the  writer  hopes  that  other  experimentalists  will  perfect  the 
theories  there  suggested.  It  is  proper  that  attention  should 
here  be  called  to  the  fact  that  our  coefficient  m  or  C,  as  used 
for  determining  the  flow  in  pipes  includes  all  resistances  to 
flow,  including  the  resistance  to  entry  into  the  pipe.  No  sep- 
arate provision  was  made  in  the  formula  for  the  resistance 
to  entry  because  it  is  a  matter  of  no  practical  consequence 
under  ordinary  circumstances,  or  in  any  case  except  for  high 
velocities  in  very  short  pipes.  (See  remarks  under  Group  No. 
1,  §  14.)  The  writer  hopes  that  the  theory  of  coefficients  and 
the  law  of  their  variation  as  herein  presented  may  contribute 
something  new  and  valuable  to  hydraulics  as  a  science. 

For  each  degree  of  roughness  of  perimeter  there  is  a  unit 
value  of  the  coefficient  from  which  unit  value  the  coefficient 
varies  as  f/Rs",  or  as  f/D^when  V=Cy~S;  or  C  varies  as 
t/~R~or  f/ D~if  we  write  V=CyRS.  At  section  20  the  var- 
ious methods  of  writing  the  formula  are  given.  As  the  old 
theories  and  formulas  are  generally  admitted  to  be  erroneous 
they  have  been  given  no  space  in  this  volume  except  in  a  few 
instances,  where  the  defects  in  the  best  of  them  have  been 
pointed  out  in  the  course  of  demonstrating  the  new  princi- 
ples herein  presented. 

Perhaps  the  great  variety  of  theories  of  variation   of   the 


8  SULLIVAN'S  NEW  HYDRAULICS. 

coefficient,  and  of  formulas  for  flow  will  beet  exhibit  the  pres- 
ent confuBed  and  uncertain  knowledge  of  hydraulics.  A  few 
of  the  leading  formulas  for  flow  in  pipes,  and  a  few  of  the 
leading  formulas  for  flow  in  open  channels  are  here  given  in 
order  to  illustrate  the  general  confusion  with  which  every 
student  of  hydraulics  has  met.  It  will  be  noted  that  all,  or 
nearly  all,  these  formulas  may  be  reduced  to  the  form  V^Cv/KS. 
Hence  the  main  difference  in  them  lies  in  the  different  theor- 
ies of  the  variation  of  C.  We  find  in  most  cases  the  same 
author  gives  a  different  law  of  variation  in  C  for  pipes  from 
that  given  for  open  channels,  as  though  the  law  depended 
upon'the  form  of  the  channel,  and  changed  with  the  change 
in  form  of  channel.  Others  adopt  a  constant  unit  value  of  C 
for  all  classes  and  degrees  of  roughness  of  both  open  channels 
and  pipes,  and  make  C  vary  with  ^/TTonly.  Hence  the  same 
value  of  i/RS  will  give  the  same  result  by  such  formulas  for  open 
channels  as  for  pipes,  and  for  rough  as  for  smooth  perimeter. 
If,  in  the  following  formulas,  the  cofficient  formula  for  C  be 
separated  from  the  formula  for  V,  the  various  theories  for 
variation  in  C  at  once  appear,  and  we  at  once  see  why  it  is 
that  scarcely  any  two  formulas  will  give  like  results  for 
the  same  conditions.  It  is  evident,  therefore,  that  if  one  is 
right  all  the  others  are  wrong. 


for  pipes (Fanning) 


.000077^+ 


RS 

00000162  for  Pips'3 (D'Arcy) 


SULLIVAN'S  NEW  HYDRAULICS. 

V=105.926  (RS)H  for  pipes (Saint  Vennant) 

V=(9579  RS  +  .00813)  *—  0.0902  for  pipes 

(D'Aubuisson) 

V=47.913  ^Sd^for  pipes (Black well) 

V=100  ^RS  for  pipes (Leslie) 


V  =•/(!  1703.95  RS  +  .01698)  —  01308  for  pipes  ....... 

.....................................  (Eytelwein) 


til 


for  pipes  ...........  (Hawkeslej  ) 


/+54d 


V=v/  (9978.76  RS  +  .02375)  —  0.15412  for  pipes  ...... 

•  ...................  .....................  (Prony) 

V=J  _  —  _  for  pi  pet-  ......  (Neville) 

A/.0234  R  +  0001085  I 

1.811      .00281 


//R/ 
channels  ...............................  (Kutter) 

In  the  Kutter  formula  n  represents  the  degree  of  rough- 
ness, and 


In  the  formula  for  n 
1=1.811;  C=-5L=;  B  = 


It  is  to  be  observed  in  regard  to  the  variation  of  Ganguil- 

let  and  Kutter's  C  that 

First — In  all  cases  of  pipes  or  open  channels,  where  the  hy- 
draulic mean  radius  (R)  is  less  than  3.281  feet,  an  in- 
crease in  slope  (S)  will  increase  the  value  of  C. 

Second— In  all  cases  of  pipes  or  open  channels  where  R  is 
greater  than  3.281  feet,  an  increase  in  S  will  cause  C 
to  decrease. 


10  SULLIVAN'S  NEW  HYDRAULICS. 

Third— Where  R=3.28l  feet  exactly,  the  value  of  C   will  be 

1      fil  1 

constant    for    all    slopes   and  will  equal— — — 

n 

At  page  106  of  Ganguillet  and  Kutters  work  (Hering 
and  Trautwine'stranslation)an  unsatisfactory  attempt 
is  made  to  explain  this  reverse  variation  of  the  coeffi- 
cient. In  the  translator's  preface  it  is  pertinently 
stated  "that  the  laws  of  flowing  water  must  be  the 
same  whether  the  channel  is  large  or  small,  slightly 
inclined  or  precipitous."  In  this  remark  the  writer 
fully  concurs.  The  above  variations  of  C  would  lead 
to  the  conclusion  that  the  law  of  gravity  reversed  it- 
self at  the  point  where  R=3  281  feet.  Such  variation 
is  clearly  erroneous  and  is  unsupported  by  any  sound 
theory  or  facts. 

In  order  that  the  various  theories  of  flow  in  open  channels 
may  be  compared  with  the  theories  of  flow  in  pipes,  and  their 
differences  noted,  a  collection  of  the  most  prominent  formu- 
las for  flow  in  open  channels  will  here  be  given: 

l/T  =  Af  R~Xt/~S~  (Gauckler.) 

In  Gauckler's  formula  A  is  supposed  to  be  constant  for 
any  given  roughness,  and  the  coefficient  varies  as  tf~R. 

V=92.20  t/RS (Brahms  &  Eytelwein) . 

In  Brahm's  formula  the  coefficient  varies  only  as  -^  R 

V=4.9  R  e/~S~for  small  streams (Hagen.) 

Here  the  coefficient  varies  directly  with  R. 
V=3.34  y~R  X  e/S~  for  large  streams. .  .(Hagen) 

Hew  the  coefficient  varies  with  y'  ~R 


(Bazin) 


In  Bazin's  [formula  A  and  B  are  constant  for  any  given 
degree  of  roughness  of  perimeter. 


SULLIVAN'S  NEW  HYDRAULICS.  11 

V=(  _  1000e  _  \l  .....  (D'Arcy  and  Bazin) 
V-08534R  +  .35  ) 

y  _   |2gRS  ............................  (Fanning) 

In  Panning's  formula  m  =     ya   •  It  is  a  direct  coefficient 

which  decreases  asV*  increases  or  as  the  roughness  decreases 
and  also  varies  with  R.  The  mean  values  of  m  for  channels  in 
earth  of  ordinary  roughness  vary  from  0.05  for  R=  0.25  to  m 
=0.002  for  R=25.00.  For  very  rough  channels  the  value  of  m 
would  be  greater  because  V*  would  decrease  as  the  rough  - 
cess  increased.  To  show  the  theory  of  this  formula  it  should 


be   written  VJL     X    V  S    and   m=x'2  g  R        The 

g 
value  of  -yg  depends  upon  the  degree  of  roughness  only  and 

varies  with  R  instead  of  /  R^  as  it  should. 

V=140v/RS—  11  f  RS  .  ....  .................  (Neville) 


V=v/ 1067.02  RS +0-0556— 0.236 (Prony) 


V=v/10567.80RS  +  267—1.64 (Girard) 


V=v/8976  50  RS  +  0.012—0.109 (D'Aubuisson) 


V=-/  8975.43  RS+0.011589  —0.1089  ......  (Eytelwein) 

V=1(KVRS  ................  (Pole,  Leslie,  Beardmore) 


^225  R  v 

(Humphreys  &  Abbot) 


In  Humphreys  &  Abbot's  formula  R    ^yet  p_L.  \yidth  an(*  m 
1.69 


~!/R  +1.  5 

It  will  be  noted  that  a  majority  of  these  formulas  make 


12  SULLIVAN'S  NEW  HYDRAULICS. 

no  provision  whatever  for  different  degrees  of  roughness 
or  different  classes  of  wet  perimeter,  and  hence  the  computed 
results  will  be  the  same  for  rough,  stony  channels  of  irreg- 
ular cross  section  as  for  smooth  channels  in  firm  earth  with 
uniform  cross  sections.  The  experimental  coefficients  devel- 
oped from  actual  guagings  of  different  clasees  of  streams,  and 
tabulated  in  the  following  pages  of  this  work  show  that  the 
value  of  C  varies  from  27  to  75  in  channels  of  like  dimensions 
and  slope,  but  of  different  degrees  of  roughness  of  wet  peri- 
meter. The  different  classes  of  perimeter  must  therefore  be 
classified  and  the  unit  value  of  C  for  each  class  must  be  es- 
tablished experimentally  by  actual  guagings.  When  the 
proper  unit  value  of  the  coefficient  for  each  degree  of  rough- 
ness is  thus  established,  it  must  thereafter  vary  correctly 
with  all  changes  of  conditions  as  to  slope,  hydraulic  mean 
depth, ete,,so  that  the  one  unit  coefficient  will  accurately  apply 
to  all  channels  in  the  same  class  of  roughness,  regardless  of 
dimensions  and  slope  or  velocity  of  flow. 

Ganguillet  &  Kutter  recognized  the  necessity  of  classify- 
ing the  degrees  of  roughness  of  wet  perimeter  andof.establish- 
ing  the  unit  value  of  th^  coefficient  for  each  class.  They 
adopted  the  unit  value  of  their  coefficient  of  roughness,  n, 
for  each  degree  of  roughness,  and  this  value  of  n  was  to  ap- 
ply to  all  perimeters  of  like  roughness,  regardless  of  the  di- 
mensions of  the  channel,  but  they  failed  to  discover  the  law 
which  governs  the  true  variation  of  n  and  hence  the  value  of 
their  C  will  not  vary  correctly  with  changing  conditions. 

Mr.  J.  T.  Panning  also  recognized  the  necessity  of  classi- 
fying coefficients  in  accordance  with  the  degree  of  roughness 
of  the  perimeter,  but  he  adopted  a  system  of  direct  coeffi- 
cients for  each  velocity  and  for  each  hydraulic  mean  depth, 
instead  of  determining  the  unit  value  for  each  class  of  peri- 
meter. The  difficulty  with  such  direct  coefficients  is  that 
they  will  apply  only  to  cases  exactly  similar  to  the  conditions 
under  which  they  were  determined.  The  result  is  that  we 
must  have  a  separate  coefficient  for  each  velocity  in  the  same 
channel,  and  for  each  change  in  hydraulic  mean  depth,  and 


SULLIVAN'S  NEW  HYDRAULICS.  13 

for  each  degree  of  roughness.  It  is  a  fixed,  inflexible  quan- 
tity whose  value  must  be  ascertained  by  experiment  for  ev- 
ery change  in  any  of  the  conditions.  When  the  student  of 
hydraulics  investigates  and  compares  the  conflicting  theories 
of  flow  and  of  the  variation  of  the  coefficient  as  set  forth  in 
the  old  formulas,  he  is  simply  bewildered  and  discouraged, 
for  he  can  discover  no  satisfactory  reason  for  adopting  any 
one  of  them  in  preference  to  another.  The  writer  therefore 
hopes  to  be  pardoned  for  offering  what  he  conceives  to  be  the 
rational  solution  of  these  difficulties. 

MARVIN  E.  SULLIVAN. 


CONTENTS. 


Introductory.--»Evolution  of  the  Formula. 
Discussion  of  Present  Data  of  Flow. 
Reverse  Variation  of  Coefficients  Explained. 

CHAPTER  I. 

SECTION    1.    The  law  of  falling  bodies. 
"          2.     The  laws  of  fluid  friction. 

CHAPTER  II. 

SECTION    3.    Properties  of  the  circle  and  of  open  channels. 
"          4.    Coefficients  of  resistance. 
"          5.     Coefficients  of  velocity. 
'          6.    The  law  of  variation  of  coefficients. 
"          7.     Deduction  of  general  formulas  for  flow. 
"          8.    Variation  in  the  coefficient  illustrated. 
'  9.    Practical  determination  of  coefficients  of  resist- 

ance. 

"        10,    Conversion  of  coefficients. 
'        11.    Determination  of  coefficients  of  velocity. 
"        12.     Value  of  coefficient  affected  by  density  of  peri- 

meter. 
"        13.     To  determine  the  value  of  the  coefficient  in  cases 

where    the  flow   is  in   contact   with   different 

classes  of  perimeter. 
14.    Tables  of  cofficients  deduced  from  data  of  flow 

in  all  classes  of  pipes,  conduits,  flumes,  canals 

and  rivers,   with   remarks  and   discussions  of 

each  group. 
"        15.    Roughness  of  wet  perimeter  defined. 

CHAPTER  III. 
SECTION  16.    General  formulas  in  terms  ot  diameter  in  feet. 


SULLIVAN'S  NEW  HYDRAULICS.  15 

SECTION  17.    General  formulas  in  terms  of  pressure,  quantity 

and  diameter. 
"        18.     General  formulas  in   terms  of  hydraulic  mean 

radius. 

"  19.  Application  and  limitation  of  formulas. 
"  20,  General  formulas  using  C  instead  of  m. 
"  21.  General  formulas  in  terms  of  hydraulic  radius 

using  C. 
"        22.    Special  formula  for  vertical  pipee. 

CHAPTER  IV. 

SECTION  23.  Table  for  ascertaining  discharge  or  velocity  in 
cast  iron  pipes.  Table  No.  2  for  discharge  of 
asphaltum  coated  pipes.  . 

"  24.  Table  for  velocity  and  discharge  of  brick  lined 
circular  sewers  and  conduits  flowing  full. 

"  25.  Egg  shaped  brick  sewers,  elementary  dimensions 
of.  Table  for  velocity  and  discharge  of  egg 
shaped  sewers. 

"        26.     Short  formulas  for  use  with  foregoing  tables. 
27.    Table  No.  5.    Value  of  d,  and  r  with  roots  and 
powers. 

"  28.  Trapezoidal  canale.  To  find  mean  radius  and 
area.  Tables  for  velocity  and  discharge  of  trap- 
ezoidal canals. 

"  89.  Rectangular  canals,  flumes  and  conduits.  Tables 
for  velocity  and  discharge. 

"  30.  Table  showing  fall  per  mile,  distance  in  which 
there  is  a  fall  of  one  foot,  together  with  values 
of  S  and  v/ST 

"  31.  Table  and  rules  for  finding  required  slope  of 
cast  iron  pipe  to  generate  any  required  velocity. 

"  32.  Head  in  feet  lost  by  friction  in  cast  iron  pipe. 
Table  No.  17  for  finding  friction  loss  for  any  ve- 
locity. 

"  33.  Formulas  and  table  for  finding  loss  by  friction  in 
any  class  of  pipe  for  a  given  discharge. 


16  SULLIVAN'S  NEW  HYDRAULICS. 

SECTION  34.  Formulae  and  table  for  friction  loss  in  asphaltum 
coated  pipe  for  any  velocity. 

"        35.    Flow  and  resistance  in  tire  hose,  with  formulas. 

"  36.  Pressure  at  hydrant  or  steamer  required  to  force 
the  discharge  of  a  given  number  of  gallons  per 
minute. 

"  37.  Friction  loss  in  fire  nozzles.  Formulas  and  dis- 
cussion. Table  No.  20,  showing  friction  loss  in 
fire  nozzles. 

"        38.    Friction  loss  in  ring  fire  nozzles. 

"  39.  Friction  loss  in  hydraulic  power  nozzles,  re- 
ducers, &c. 

"  40.  Multipliers  for  finding  friction  loss  in  cast  iron 
giants. 

"  42.  Total  head  or  slope  of  cast  iron  pipe  to  cause 
given  discharge. 

"  .     43.    To  find  diameter  of  pipe  required. 

"  44.  Loss  by  friction  in  cast  imn  pipe  for  given  dis- 
charge. 

'•  45.  Table  of  multipliers  for  finding  friction  loss  in 
cast  iron  pipe  for  any  given  discharge. 

"  46.  Friction  loss  for  given  discharge  of  asphaltum 
coated  pipe.  Table  No.  23  for  friction  loss  per 
100  feet  length. 

"  47.  To  find  the  discharge  from  the  amount  of  fric- 
tion lose. 

"        48.    To  find  the  discharge  from  slope  and  diameter. 

11  49.  Power  at  pump  required  to  force  a  given  dis- 
charge. 

"        50.    To  find  the  discharge  from  the  pressure. 

"  51.  Lbs.  pressure  lost  by  friction  for  a  given  dis- 
charge. 

"  52.  Slope  required  to  cause  a  given  discharge.  Ta- 
ble No.  24  for  finding  any  required  slope. 

CHAPTER  IV. 
SECTION    53.    Wooden  stave  pipe. 


SULLIVAN'S  NEW  HYDRAULICS.  17 

SECTION  54.  Wooden  pipes  compared  with  other  classes  of 
pipe. 

"         55.    Earthware  or  vitrified  pipe. 

"         56.    Table  No.  56,  elementary  dimensions  of  pipes. 

"  57.  Length  in  feet  of  small  pipes  to  hold  one  U.  S 
gallon. 

"  58.  Decimal  equivalents  to  fractional  parts  of  a 
lineal  inch.  Fractional  inches  in  equivalent 
decimals  of  a  foot.  Tenths  of  a  foot  in  equiva- 
lent inches. 

"  59.  Conversion  tables  of  weights  and  measures  for 
converting  U.  S.  to  metrical  and  metrical  to  U. 
S.  measures. 

CHAPTER  V. 

SECTION  60.  Work,  power  and  horse-power  defined.  Form- 
ulas for  finding  the  horse-power  of  a  stream. 

"  61.  To  find  cubic  feet  of  water  required  to  gener- 
ate a  given  power. 

"         62.    To  find  the  net  head  required  to  generate  a 
given   power.    Efficiency  of  water   wheels  de- 
fined. 
63.    Head  of  water  defined. 

"  64.  To  find  the  diameter  of  a  pipe  which  will  carry 
a  given  quantity  of  water  with  a  given  loss  of 
head  by  friction. 

"  65.  To  find  the  required  area  and  diameter  of  noz- 
zle to  discharge  a  given  quantity. 

"         66,    Pipe  lines  of  irregular  diameter. 

"  67.  A  power  main  with  nozzle  and  water  wheel  to 
run  at  a  given  speed  and  develop  a  given  power. 

"  68.  Table  No.  36.  Eleventh  roots  and  powers. 
Converse  Lock  Joint  pipe.  Remark  3. 

"         69.    Friction  loss  at  bends  in  pipes;  formulae. 

"  70.  Weisbach  and  Rankine's  formulas  for  friction  at 
bends. 

"         71.    Results  of  different  formulas  compared. 


18  SULLIVAN'S  NEW  HYDRAULICS. 

SECTION    72.    Resistance  at  bends.    Ronnie's  experiments. 

"  73.  Thickness  of  pipe  shell  proportional  to  diame- 
ter and  pressure. 

"         74.    Value  of  S  in  water  pipe  formulas. 

"         75.     Riveting  and  riveted  pipe. 

76.     Proportions  of  single   and  double   riveted  lap 
joints. 

"  77.  Table  of  decimal  equivalents  to  fractional 
parts  of  an  inch. 

"  78.  Weight  and  thickness  of  sheet  iron  and  sheet 
steel. 

"          79.    Rules  for  finding  weight  of  riveted  pipe. 

"          80.     Tests  for  strength  of  lap  riveted  joints. 

"          81.     Testing  iron  and  steel  plates  for  defects. 

"         82.     Pipe  joints — how  made. 

"  83.  Formula  for  proportions  of  reducers  for  joining 
large  to  small  pipes. 

CHAPTER  VI. 

SECTION  84.  Flow  in  open  channels — permanent  and  uniform 
flow. 

"  85.  Resistances  and  net  mean  head  in  open  chan- 
nels. 

"         86.    Ratio  of  surface,  mean  and  bottom  velocities. 

"          87.     The  eroding  velocity. 

"  88.  Eroding  velocity  in  straight  canals  of  uniform 
section. 

"  89.  Slope  required  to  generate  given  bottom  veloc- 
ity. 

"         90.    Stability  of  channel  bed. 

"  91.  Adjustment  of  grade,  bottom  velocity  and  side 
slopes. 

"  92.  Dimensions  of  canals  to  discharge  given 
quantities. 

"  93.  Allowance  in  cross-eection  of  canal  for  leakage 
and  evaporation. 

"         94.     Flume  forming  part  of  canal. 

"         95.    Float  measurement  of  mean  velocity  in  canals. 


SULLIVAN'S  NEW  HYDRAULICS.  19 

APPENDIX  I. 

SECTION  96.     Weirs  and  weir  coefficients.    Discussion. 

APPENDIX  II. 
SECTION  97.     Water  works  rules  and  data. 

"         98.    Quantity  of  water  required  per  person. 

"         99.     Consumption  and  cost  per  1,000  gallons  in  var- 
ious cities. 

"        100.    Formulas  and  tables  for  diameter  of  pipe  re- 
quired. 

"        101.    Formulas  for  conduits,  pipes,  etc. 

'•       102.    Formulas  for  diameter  to   maintain    a  given 
pressure  while  discharging. 

"       103.    Friction  heads,velocities  and  discharge  for  given 
diameters  and  slopes. 

•'        104.    Formulas  for  thickness  and  weight  of  cast  iron 
pipe. 

"        105.    Weight  and  dimensions  of  cast  iron  pipe  made 
by  the  Colorado  Fuel  and  Iron  Company. 

"        106.    Weight  per  foot  length  of  cast  iron  pipe  for 
150  and  200  pounds  pressure. 

"        107.     "Standard"  cast  iron  pipe. 

108.     Cost  per  100  feet  of  pipe   and  laying  in  Denver, 
Colorado. 

"        109.    Cost   per  foot  of  pipe  and  laying  in    Boston, 
Massachusetts. 

"       110.    Cost  of  trenching,  laying,  calking,  etc.,  in  Om- 
aha, Nebraska. 

"        111.     Weston's  table  for  estimating  cost  of  pipe  lay- 
ing. 

112.  Table  showing  cubic  yards  of  excavation  in 
trench  per  foot. 

113.  Bell-holes  in  pipe  trench. 

"       114.     Proper  depth  of  trench  for  pipe  laying. 

"        115.     Amount  of  trenching  and  laying  per  man  per 

day. 
"       116.    Amount  of  lead  required  per  joint  for  cast  iron 

pipes. 


"There  is  in  this  world  but  one  work  worthy  of  a  man, 
the  production  of  a  truth,  to  which  we  devote  ourselves,  and 
in  which  we  believe." — Taine. 


INTRODUCTORY. 


The  evolution  of  the  formula  and  discussion  of  the  present 
available  data  of  flow,  with  an  explanation  of  the  reverse 
variation  of  coefficients, 


The  general  formula  for  flow  as  herein  finally  presented 
may  justly  be  called  the  result  of  the  combined  labors  of 
Galileo  and  all  subsequent  writers  and  experimentalists,  in- 
cluding the  present  writer.  The  present  writer  has  accepted 
and  adopted  from  all  former  writers  on  hydraulics  such  prin- 
ciples and  theories  as  have  been  thoroughly  proven  true  and 
general,  and  has  rejected  all  theories  of  doubtful  or  uncertain 
value  and  supplied  the  deficiencies  thus  arising  by  original 
investigations  and  experiments.  The  foundation  of  the  form- 
ula was  the  discovery,  early  in  the  seventeenth  century,  of 
the  law  of  falling  bodies  by  Galileo.  In  his  investigations  of 
flow  in  rivers  Galileo  failed  to  recognize  the  nature  of  the 
resistance  of  the  solid  wet  perimeter  and  the  difference  be- 
tween the  resistance  of  a  solid  in  contact  with  a  liquid,  and 
that  of  two  solid  bodies  in  contact.  His  investigations  there- 
fore resulted  in  failure.  Later  it  was  discovered  by  Torri- 
celli,  a  pupil  of  Galileo,  that  resistances  aside,  the  square  of 
the  velocity  is  directly  proportional  to  the  head  or  inclination, 
or  in  other  words,  that  the  velocity  would  be  as  the  square 
root  of  the  head  or  elope. 

Brahms  discovered  that  the  acceleration  which 
would  occur  according  to  the  law  of  gravity  did  not  actually 
occur,  but  that  the  velocity  of  flow  became  constant.  His 
investigations  established  the  fact  that  the  solid  wet  peri- 


22  SULLIVAN'S  NEW  HYDRAULICS. 

meter  offered  a  resistance  to  the  flow  which  opposed  the  ac- 
celeration that  would  otherwise  occur,  and  he  assumed  th»t 
this  resistance  was  directly  proportional  to  the  hydraulic 
mean  radius,  or  to  the  area  of  cross-section  of  the  column  of 
water  divided  by  the  wet  girth.  In  the  latter  part  of  the 
eighteenth  century,  Du  Buat  instituted  a  series  of  experi- 
ments from  the  results  of  which  he  discovered  that  the 
velocity  of  flow  depended  upon  the  slope  of  the  water  surface 
or  head,  and  that  in  channels  of  uniform  area  and  grade, 
when  equilibrium  was  attained,  the  flow  became  uniform  and 
the  resistance  equalled  the  acceleration  of  gravity.  Thus 
each  investigator  has  contributed  some  valuable  discovery  or 
fact  which  has  been  able  to  stand,  while  many  of  their  as- 
sumptions have  been  proven  wholly  erroneous. 

Du  Buat  also  discovered  that  the  resistance  of  a  solid  in 
contact  with  a  liquid,  was  in  no  manner  increased  or  de- 
creased by  a  change  of  pressure  between  the  liquid  surface 
and  the  solid  surface.  In  other  words  he  discovered  that 
the  pressure  with  which  a  liquid  is  pressed  upon  a  solid  does 
not  affect  the  friction  between  them.  Du  Buat  and  Prony 
each  discovered,  as  a  consequence  of  the  law  of  gravity,  that 
the  head  or  slope  had  no  influence  whatever  upon  the  value 
or  variation  of  the  coefficient,  but  they  erroneously  assumed 
also  that  the  character  of  the  wet  perimeter  had  no  influence 
upon  the  coefficient.  It  was  the  opinion  of  Du  Buat  (and 
adopted  by  Prony)  that  the  nature  of  the  walls  and  bottom  of 
a  channel  could  not  affect  the  coefficient  because,  as  Du  Buat 
observed,  "A  layer  of  water  adheres  to  the  walls,  and  is  there 
fore  to  be  considered  as  the  wall  proper  which  surrounds  the 
flowing  mass."  With  this  view,  he  supposed  all  perimeters 
to  be  practically  "water  perimeters",|and  consequently  equally 
smooth.  It  remained  for  D'Arcy  and  Bazin  to  demonstrate 
by  many  practical  experiments  that  this  latter  assumption  of 
Du  Buat  and  Prony  was  entirely  without  foundation. 

As  a  result  of  the  experiments  of  D'Arcy  and  Bazin  the 
fact  was  established  that  the  coefficient  of  flow  would  vary 
directly  as  the  degree  of  roughness  or  smoothness  of  wet 


SULLIVAN'S  NEW  HYDRAULICS.  23 

perimeter.  Bazin  stated  also  that  the  coefficient  varied  with 
the  value  of  the  hydraulic  mean  radius,  thus  confirming  the 
observations  of  Brahms.  D'Arcy  and  Bazin  recognized  that 
the  slope  or  head  had  no  influence  upon  the  value  or  variation 
of  the  coefficient,  and  hence  omitted  that  feature  in  their 
formula,  and  assumed  with  Brahms  that  the  total  resistance 
for  any  given  degree  of  roughness  would  be  directly  propor- 
tional to  R.  the  hydraulic  mean  radiue.  They  were  correct 
in  their  assumptions  thus  far,  but  they  failed  to  go  one  step 
farther  and  provide  for  the  acceleration  as  well  as  the  re- 
sistance, or  in  other  words  to  ascertain  the  mean  resistance  of 
all  the  particles  of  the  entire  cross-section  by  taking  the 
product  of  total  retardation  by  total  acceleration.  They 
adopted  and  embodied  in  their  formula  simply  the  feature 
of  total  retardation  without  modification  by  the  acceleration. 
Hence  they  failed  to  ascertain  the  mean  resistance  of  the 
entire  cross-section,  and  as  a  necessary  result  of  adopting 
total  resistance  instead,  their  formula  gives  results  too  low 
in  large  pipes  or  channels  and  the  larger  the  pipe  or  channel, 
the  greater  the  error  will  become. 

Ganguillet  and  Kutter  proposed  a  formula  based  partly 
on  Bazin's  formula  and  partly  upon  the  results  of  some  ill 
assorted  guagings.  While  this  latter  formula  has  become 
popular  and  is  considered  as  standard  by  many  engineers,  it 
is  really  based  upon  theories  which  are  directly  contradic- 
tory of  both  the  laws  of  friction  and  of  gravity,and  a  short  in- 
vestigation will  expose  the  fact  that  it  can  be  applied  with 
accuracy  only  to  open  channels  of  very  slight  inclination,  and 
whose  mean  radii  approach  closely  to  unity.  The  discussion 
of  coefficients  will  point  out  the  reasons  why  this  is  true. 

The  writer  would  also  remark  that  the  published  tables 
of  data  in  relation  to  flow  in  pipes  and  open  channels  are,  in 
a  large  majority  of  cases, wholly  unreliable,  as  many  contract- 
ing engineers  have  recently  discovered  to  their  great  cost. 

Coefficients  should  never  be  based  upon  data  of  uncer- 
tain value,  as  the  results  must  depend  upon  the  correctness 
of  the  coefficient  used. 


24  SULLIVAN'S  NEW  HYDRAULICS. 

The  data  of  guagings  of  rivers  at  different  stages  and  for 
various  depths  of  flow  are  nearly  all  worthless  for  scientific 
purposes  for  one  or  more  of  the  following  reasons: 

1 .  The  data  fail  to  show  whether  the  stream  was  rising 
or  falling  or  stationary  when  the   mean  velocity  and 
corresponding  alope    of  water    surface   were  ascer- 
tained. 

Where  a  stream  is  either  rising  or  falling  with 
considerable  rapidity,  there  is  little  or  no  relation  be- 
tween the  slope  of  the  water  surface  and  the  actual 
mean  velocity  then  prevailing.  The  same  depth  of 
flow  or  guage  height  at  the  same  point  does  not 
necessarily  always  produce  the  same  slope  of  water 
surface.  The  slopes  are  usually  thus  recorded  as 
corresponding  to  a  certain  guage  height  without 
actual  measurement.  The  same  slope  and  guage 
height  on  a  rising  river  will  cause  a  much  greater  ve- 
locity and  give  a  higher  coefficient  of  flow  than  upon 
a  falling  river.  The  difference  in  value  of  the  veloc- 
ity of  flow  and  of  the  coefficient  will  depend  upon  the 
magnitude  of  the  freshet — the  distance  it  extends  up 
stream — the  suddenness  and  rapidity  of  the  rise  or 
fall.  In  a  rising  or  falling  stream  equilibrium  is  lost 
and  the  actual  effective  slope  which  is  generating  the 
velocity  at  such  times  is  very  different  from  the  ap- 
parent slope,  and  can  be  ascertained  only  from  the 
mean  velocity  actually  existing  at  the  time,  and  from 
a  previous  knowledge  of  the  degree  of  roughness  of 
the  stream  in  that  locality.  The  effective  slope  may 
then  be  found  by  formula  for  S. 

2.  Guaging  stations  are  always   located  at  narrow,  deep 
sections  of  the  stream,  and  the  hydraulic  radius  thus 
roughly  measured  is  given   as  the    mean    hydraulic 
radius  of  the  stream.    This    is    never    the    true  nor 
even  scarcely  approximate,   mean  hydraulic  radius, 
except  at  the  particular  point  where  measured. 


SULLIVAN'S  NEW  HYDRAULICS.  25 

3.  In  many  cases  the  mean  velocities  tabled  are  deduced 
from  the  surface  velocities  by  some   absurd  formula 
which  is  basad  upon  the   theory  that  there  is  a  con- 
stant ratio  between  the   maximum  surface  velocity 
and  the  mean  velocity,  and  that  this  ratio  is  the  same 
in  all  classes   and  dimensions  and   degrees  of  rough- 
ness of  channels. 

4.  The  general  slope  is  usually  taken  and  is  assigned  as 
the  local  slope.     They   are  usually  very  different  ex- 
cept at  extreme  high  water,  when   the  general  and 
local  slope   are  nearly  equal. 

5.  In  turbulent   streams,  or  in   very  large  streams  it  is 
impossible,  for  many  reasons  to  ascertain  the  slope  of 
water  surface,  especially  the  high  water   slope.    The 
slopes  assigned  in  such  cases  are  simply  the  record 
of  a  guess,  and  have  no  value  for  scientific  purposes. 

6.  The  method   of   ascertaining    the   mean    velocity  as 
finally    tabled,    is    frequently    by  mid-depth  floats. 
These  floats  vary  in  the  time  of  passing  over  the  same 
course  by  as  much  as  25   per  cent,  depending   upon 
the  number  of  whirls,    boils    and    cross-currents  en- 
countered.   The  mean  is  taken  as   the    actual  mean 
velocity,  or  as  bearing  a  given  fixed  ratio  thereto,  re- 
gardless of  formation   of  the  channel  bed.    The  in- 
equalities of  the  bottom  of  the  stream  make  it  impos- 
sible to  adjust  a  float  to  mid  depth.     If  the  mid  depth 
velocity  were    absolutely    known,  it    is    not   known 
what  relation  it  bears  to  the    actual    mean    velocity. 
The  mean  velocity  may  be  either  above  or  below  mid- 
depth.    That  will  depend  upon    the    magnitude  and 
roughness  of  the  channel  at  the  given  place. 

7.  Many  rivers  are  affected  by  gulf   tides,  as  the  Missis- 
sippi river  as  far  up  as  Donaldsonville,  and  the  river 
Seine  at  Poissy,  Triel  and  Meulon,  where  the   water 
surface  fluctuated  as  much  as    two    feet  during  the 
time  of  the  guagings  there.     This   is   so   marked  on 


36  SULLIVAN'S  NEW  HYDRAULICS, 

the  Mississippi  river  at  Carrollton,  La.,  as  to  actually 
produce  reverse  slopes,  as  noted  by  the  river  engi- 


8.  Data  are  frequently  published  of  the  gaugings  of  a 
channel  at  different  stages  where  the  value  of  the 
hydraulic  mean  radius  increases  four  or  five  hundred 
per  cent.,  and  the  discharge  increases  by  a  thousand 
per  cent,  or  more,  and  yet  the  same  slope  is  assigned 
for  all  stages!  As  an  example  of  this  kind,  see  the 
ten  gaugings  of  the  Saone  under  the  direction  of  M. 
Leveille,  1858-9.  Here  the  hydraulic  radius  varies 
from  R=3.88  to  R=15.83,  and  yet  the  slope  of  water 
surface  is  given  as  S=.00004  for  each  of  the  ten 
gaugings.  The  gaugings  of  the  Weser  by  Funk  are 
of  no  value.  Bazin  remarks;  '-It  is  to  be  noted 
that  Funk  has  almost  always  adopted  the  same 
slope  for  an  entire  group  of  experiments."  Bazin  also 
states  that  in  the  experiments  of  Brunning  on  the 
Rhine,  "the  slopes  were  not  measured  at  all,  but 
subsequently  computed  so  as  to  make  the  results 
accord  with  the  formula." 

It  is  well  known  that  the  gaugings  of  the  Mississippi 
River  under  the  direction  of  Humphreys  and  Abbot 
during  the  year  1858  are  of  very  doubtful  value.  The 
areas  had  been  measured  at  the  gauging  stations 
seven  years  before,  and  were  assumed  to  have  remain- 
ed constant  ever  afterward,  when  in  fact  the  area  at 
a  given  point  in  that  river  is  frequently  altered  by  as 
much  as  14,000  square  feet  within  twenty-four  hours 
by  scour.  In  1858  the  velocities  were  taken  at  a 
depth  of  only  five  feet  below  surface,  and  the  mean 
velocities  were  calculated  by  an  empirical  formula  of 
no  value.  Du  Buat's  mean  velocities  as  tabled  for 
the  Canal  du  Jard,  were  deduced  from  surface  float 
velocity  by  Du  Buat's  formula.  Du  Buat's  formula 
for  ascertaining  the  mean  from  the  surface  velocity 
was  based  on  his  experiments  on  a  very  small  wooden 


SULLIVAN'S  NEW  HYDRATLICS.  27 

trough  of  smooth  perimeter,  and  has  long  since  been 
discarded  as  being  of  no  value. 

If  the  present  available  tables  of  data  were  assorted 
carefully  and  the  worthless  were  rejected,  there 
would  little  remain.  These  remarks  are  made  here 
in  order  to  call  attention  to  the  need  of  obtaining 
new  and  accurate  data,  and  to  prevent  too  great 
reliance  upon  the  value  of  such  data  as  are  now 
available.  The  data  relating  to  flow  in  pipes  and 
conduits  are  equally  bad  and  untrustworthy.  The 
data  relating  to  asphaltum  coated  pipes  except  those 
of  Hamilton  Smith  Jr.,  and  to  wrought  iron  pipes 
are  especially  of  uncertain  value,  and  no  expensive 
enterprise  should  be  based  upon  them  without  ad- 
ditional experiment.  Some  of  the  more  recent  data 
relating  to  flow  at  different  depths  in  large  masonry 
conduits  of  comparatively  short  length,  show  by  the 
value  of  the  slope  of  water  surface  as  compared  with 
the  slope  of  the  bottom  of  the  conduit,  that  there 
would  have  been  no  water  in  the  upper  end  of  the 
conduit  at  all.  It  is  probable  that  equilibrium  had 
not  bem  attained  at  the  time  of  gauging.  Any 
other  explanation  renders  the  data  absurd,  and  this 
explanation  renders  them  worthless.  In  a  channel 
of  uniform  grade,  croes-section  and  roughness  of 
perimeter,  the  slope  of  the  water  surface  will  be  the 
same  as  the  slope  of  the  channel  bed  as  soon  as  uni- 
form flow  is  established.  If  this  were  not  the  case, 
uniform  flow  could  never  occur,  because  the  water 
would  be  of  greater  depth  at  one  point  than  at 
another,  and  the  velocities  would  be  inversely  as  the 
depths  or  wetted  areas.  An  investigation  of  the  data 
of  flow  which  is  now  available  is  discouraging  to  a 
degree.  It  is  a  misfortune  common  to  us  all.  In 
large  rivers  where  the  roughness  of  perimeter  and 
the  area  of  cross-section  vary  at  almost  every  foot  in 
length,  and  where  scour  or  fill  is  constantly  in  pro- 


SULLIVAN'S  NEW  HYDRAULICS, 

gross,  it  is  impossible  that  equilibrium  in  its  true 
sense,  should  ever  be  established.  The  flow  is  alter- 
nately checked  by  rough  perimeter  and  increased 
area  due  to  scour,  and  accelerated  by  reaches  of 
straight,  smooth  perimeter  where  the  area  is  con 
tracted.  The  flow  is  similar  to  that  in  a  compound 
pipe  made  up  of  lengths  alternately  large  and  small, 
and  alternately  smooth  and  rough.  The  velocity  is 
necessarily  inversely  as  the  areas,  in  case  the  supply 
of  water  is  constant. 

For  these  reasons  the  local  slope  over  a  very  short 
length  of  channel,  at  normal  stages  of  the  stream,  is 
the  slope  that  must  be  used  in  the  application  of  a 
slope  formula.  Otherwise  the  result  by  formula  will 
be  of  no  value. 

The  value  of  the  coefficient  will  usually  decrease  with 
increase  in  depth  of  flow  at  any  given  point  along  a 
natural  channel,  after  the  depth  exceeds  the  usual 
d«pth  of  flow.  This  is  due  to  the  simple  fact  that 
the  bed  is  silted  and  worn  smoother  up  to  the  depth 
of  ordinary  flow  than  the  sides  above  the  usual  flow. 
Hence  as  depth  of  flow  increases  the  proportion  of 
the  rougher  side  wall  perimeter  increases  also,  and 
thereby  decreases  the  value  of  the  coefficient  as  depth 
of  flow  and  ratio  of  rough  perimeter  increase.  It 
frequently  occurs,  however,  that  the  reverse  of  this 
is  true,  as  for  example  in  channels  having  very 
rough,  stony  bottoms  and  comparatively  regular 
side  walls,  In  this  latter  case  the  coefficient  will 
increase  as  depth  of  flow  increases  because  with 
each  gain  in  depth  there  is  a  gain  of  the  smooth 
over  the  rough  perimeter,  and  the  mean  of  the 
roughness  of  the  perimeter  considered  as  a  whole 
becomes  less  and  less  at  each  successive  increase  of 
depth.  In  either  class  of  channels  the  value  of  the 
coefficient  must  vary  directly  with  the  mean  of  the 
roughness  of  the  entire  wet  perimeter  taken  as  a 


SULLIVAN'S  NEW  HYDRAULICS.  29 

whole.  It  therefore  follows  that  if  the  whole 
perimeter  be  of  equal  and  uniform  roughness  or 
smoothness  throughout,  the  coefficient  will  remain 
absolutely  constant  for  all  depths  of  flow. 

Large  masonry  conduits,  not  being  adapted  to 
withstand  pressure  from  within,  are  built  on  small 
grades  or  inclinations  and  given  free  discharge.  It  is 
nearly  always  found  that  the  coefficient  in  such  con- 
duits is  greater  for  very  small  depths  of  flow  than  for 
greater  depths.  This  is  explained  by  the  fact  that 
much  cement  or  mortar  is  dropped  upon  the  invert  or 
bottom  during  construction  and  is  ground  into  the 
pores  and  joints  of  the  brick  by  the  tramping  of  the 
masons.  The  floor  is  worn  smooth  by  reason  of  this, 
and  the  slight  inclination  of  the  conduit  and  low  ve- 
locity permit  of  the  deposit  of  a  very  fine,  dense  silt 
upon  the  bottom  which  settles  in  and  fills  up  all  the 
irregularities  and  depressions  along  the  bottom — thus 
presenting  a  smooth,  uniform,  contiguous  bottom  peri- 
meter to  the  flow.  The  side  walls,  although  of  the 
same  material,  are  much  rougher  than  the  invert  or 
bottom,  because  the  rough  projections  of  sand  along 
the  sides  of  the  brick  are  not  rubbed  off,  and  the 
pores  and  small  cavities  are  not  filled  and  plugged  by 
mortar  tramped  in  under  pressure,  and  by  the  deposit 
and  settlement  of  fine  silt,  as  occurs  on  the  bottom. 
This  is,  however,  not  true  of  open  canals  with  paved 
bottoms  and  masonry  side  walls  where  the  slope  is 
sufficiently  great  to  generate  a  velocity  at  the  bottom 
sufficient  to  prevent  the  deposit  of  silt  or  to  scour  out 
the  joints  of  the  masonry  floor.  In  this  latter  case 
the  bottom  has  no  advantage  of  the  sides  so  far  as  re- 
lates to  roughness,  unless  it  is  better  constructed,  or 
is  composed  of  smoother  material,  or  is  in  better  re- 
pair than  the  side  walls.  In  any  given  case  the  value 
of  the  coefficient  will  be  directly  as  the  mean  rough- 
ness of  the  entire  wetted  portion  of  the  perimeter,  or 


30  SULLIVAN'S  NEW  HYDRAULICS, 

as  the  ratio  of  smooth   to    rough    perimeter    as    the 

depth  of  the  flow  varies. 

With  these  general  introductory  remarks  upon  the  evolu- 
tion of  the  formula  for  flow,  the  uncertain  value  of  the  present 
available  data  of  flow,  and  the  causes  of  contrary  variation  in 
the  value  of  the  coefficient,  the  reader  will  be  better  prepared 
to  understand  what  follows. 


CHAPTER  I. 


Of  the  Laws  of  Gravity  and  the  Laws  of  Friction  Between  a 
Liquid  and  a  Solid. 

/.  The  Law  of  Falling  Bodies— Ae  the  law  of  falling  bod- 
ies, or  of  gravity,  and  the  law  of  friction  between  a  liquid  and 
a  solid  include  all  the  elements  of  the  flow  of  water,  it  is  of 
prime  importance  that  these  laws  should  never  be  lost  sight 
of  in  any  investigation  or  application  of  hydraulic  formulas. 
The  moment  that  a  theory  or  a  formula  deviates  from  the  re- 
quirements of  any  one  of  these  natural  laws,  that  moment  it 
must  fail.  These  laws  must  be  observed  in  their  entirety. 
No  provision  must  be  either  excluded  or  violated.  The  pen- 
alty is  certain  failure  to  the  extent  of  the  evasion  or  violation. 

Let  g=feet  per  second  by  which  gravity  will  accelerate 
the  descent  of  a  falling  body.  g=32.2  at  sea  level. 

2g=64.4. 

v  =velocity  in  feet  per  second. 

H=height  in  feet,  total  fall  in  feet,  or  total  head  in  feet. 

t=time  in  seconds. 

A  body  falling  freely  from  rest  will  descend  16.1  feet  in 
the  first  second  of  time,(t)  and  will  have  acquired  a  velocity,  at 
the  end  of  the  first  second,  of  32.2  feet  per  second,  and  will 
be  accelerated  in  each  succeeding  second  32.2  feet,  so  that  for 
each  additional  second  of  time  consumed  in  falling,  there  will 
be  a  gain  in  the  rate  of  descent  equal  to  32.2  feet.  At  the  end 
of  the  first  second  the  rate  of  velocity  will  be  32.2  feet  per 
second,  at  the  end  of  the  second  second  of  time,  the  rate  of 
velocity  will  be  64.4  feet  per  second,  and  BO  on  for  any  num- 
ber of  seconds,  adding  32.2  feet  to  the  rate  of  velocity  for  each 
second  of  time. 

Velocity  is  the  rate  of   motion.     Acceleration  is  the  gain 


32  SULLIVAN'S  NEW  HYDRAULICS. 

in  this  rate.  The  acceleration  or  gain  in  rate  of  motion  in 
the  case  of  a  body  falling  freely,  is  32.2  feet  per  second,  and 
consequently  the  velocity,  (v)  at  any  time  (t)  is  equal  to  gXt, 
or  to  the  acceleration  per  second  (g)  multiplied  by  the  num- 
ber of  seconds  of  time  (t).  The  distance  in  feet  (H)  fallen 
through  by  a  body  in  the  first  second  is  16.1  feet,  or  one  half 
g,  and  the  distance  fallen  in  any  given  time(t)is  as  the  square 
of  that  time  (t8).  Consequently  the  velocities  are  as  the 
square  roots  of  the  distances  or  vertical  heights  fallen 
through,  or  as  -j/  H.  As  gravity  produces  the  velocity  of  2 
in  falling  through  the  height  1,  the  height  in  feet  fallen  mul- 
tiplied by  2g  will  equal  the  square  of  the  velocity  in  feet  per 
second  or 

v*=2gH (1) 

From  this  fundamental  law  of  gravity  we  have 


v=-/2  gH=/  64.4H  =8.025V/  H  ..........  (2) 


v=gt  ......................................  (4) 

g=I  =JZl=32.2  at  sea  level  ................  (5) 

t       2H 


.........  (6)      1 

The  velocity  head,  or  that  portion   of   the  head  which  is 
producing  the  velocity  of  flow  in  any  case  is  therefore 

-=r  .......  .....  ;  .......................  (7> 

And  the  velocity  generated  by  any  velocity  head  is  equal 
v=  /2g    hv=8.025  y  "hv  ....................  (8) 

2.     The  Laws  of  Friction  as  Applied  to  a  Liquid  in  con- 
tact with  a  Solid  Surface—  The  results  of   many  very  careful 
experiments  establish  the  correctness  of  the  following  rules: 
I.    The  friction  on  auy  given  unit  of  surface  will  be  directly 


SULLIVAN'S  NEW  HYDRAULICS.  33 

as  the  roughness  or  smoothness  of  that  surface. 

II.    The  total  resistance  will  be  as  the  total  number  of  unite 

of  friction  surface. 

III.  The  friction  on  any  given  unit  of  surface  will  be  aug- 
mented as  the  square  of  the  velocity  with  which  the 
liquid  is  impelled  along  that  surface. 

IV.  The  friction  between  the  molecules  or  particles  of  the 
liquid  themselves,  is  infinitely  small,  and  may  be  en- 
tirely neglected. 

V.  The  friction  between  a  liquid  and  a  solid  is  not  affected 
by  the  pressure  with  which  the  liquid  is  pressed  per- 
pendicularly upon  the  solid.  The  friction  is  entirely 
independent  of  the  amount  of  radial  pressure. 

VI.  The  mean  resistance  of  all  the  particles  of  the  entire 
cross  section  of  the  liquid  vein  considered  as  a  whole, 
will  be  as  the  total  retardation  or  loss  of  velocity  by  re- 
sistance, as  modified  by  the  total  acceleration  or  free 
and  unretarded  flow,  or  as  the  product  of  total  retard- 
ation by  total  acceleration.  Total  acceleration  will  be 
as  the  square  root  of  the  net  free  head.  Total  retarda- 
ation  will  be  as  the  square  root  of  the  head  consumed 
or  loet  by  resistance. 

The  mean  resistance,  or  mean  loss  of  head,  of  all  the 
particles  of  the  entire  cross  section  taken  as  a  whole 
will  be  as  the  product  of  total  retardation  by  total  ac- 
celeration. 

The  mean  velocity  of  all  the  particles  in  a  cross-section 
will  be  as  the  square  root  of  the  mean  head  of  all  the 
particles. 


CHAPTER  II. 

Of  Coefficients  and   their  Variation- 


3.  Properties  of  the  Circle— IK  order  to  exhibit  the 
properties  of  the  circle,  and  the  relations  of  area  to  friction 
surface  in  both  open  and  closed  channels,  and  the  relations 
common  to  both  open  channels  of  any  form  and  to  circular 
closed  channels  or  pipes,  and  to  also  show  the  relation  of 
theee  common  properties  to  the  value  and  variation  of  the 
coefficients,  the  following  tables  of  circles  and  of  open  chan- 
nels of  various  forms  will  he  referred  to.  The  notation  here 
given  will  be  followed  throughout: 

H  =  total  head  in  feet. 

h"=friction  head,  or  head  required  to  balance  the  total 
resistence. 

hv  =velocity  head  in  feet  in  the  total  length  /. 

Z=1ength  in  feet  of  pipe  or  channel. 

v=mean  velocity  in  feet  per  second. 

d=diameter  in  feet. 

a=area  in  square  feet. 

p=wet   perimeter  in  lineal  feet,  or  friction  surface. 

r=— ^hydraulic  mean  radius  in  feet. 
n=coefficient  of  friction  or  of  resistance. 
m=ccefficient  of  flow  or  of  velocity. 

total  head   in  feet     H 

S=total  length  in  feet=  r=8ine  of  «loPe=to*al  head  P«* 
foot. 

h" 
S'=— =friction  head  per  foot  length  of  channel  or  pipe 

Sy  =  Velocity  head  per  foot  length  of  pipe  or  channel. 


SULLIVAN'S  NEW  HYDRAULICS, 


35 


In  full  pipes  or  circular  closed   channels  r=—  =  d  X  -25, 
and  d  =4r.     a=dsX.7854,  or  a=(4r)»X-7854=r8Xl2.5664. 
TABLE  OF   CIRCLES. 


d 

FEET 

v/d 

FEET 

r 

FEET 

a 

SQ    FEET 

P 

FEET 

BELATI'N 
OF  p  to  a 

0.50 
1.00 
2.00 
4.00 
8.00 
16.00 
32.00 

0.707 
1.000 
1.4142 
2.000 
2.828 
4.0CO 
5.657 

0.125 
0.25 
0.50 
1.00 
2.00 
4  00 
8.00 

0.1C635 
0.78o4 
3.1416 
12.5664 
50.2656 
201.0624 
804.2196 

1.5708 
3.1416 
6.2832 
12.5664 
25.1328 
50.2656 
100.5312 

p=8a 
p=4a 
p=2a 
p=  a 
p=y,a 
p=&a 
P=Ha 

It  will  be  observed  that  the  diameter  d,  is  doubled  here 
each  time,  and  that  the  result  of  doubling  d  is  to  also  double 
r  and  p.  It  follows  therefore  that  in  circular  closed  channels 
and  pipes  d,  r  and  p  vary  in  the  same  ratio.  As  d,  r  and  p 
all  vary  exactly  in  the  rame  ratio,  and  as  the  area  varies  as  da 
or  r*,  and  as  d  or  r  must  therefore  vary  as  the  square  root  of 
the  area,  it  follows  that  the  friction  surface  p,  must  also  vary 
as  the  square  root  of  the  area.  It  will  be  observed  that  when 
d,  r  or  p  is  doubled,  the  area  is  increased  four  times.  Hence  if 
the  friction  surface  p  is  doubled  the  area  is  increased  four 
times.  The  right  hand  column  of  the  table  shows  how  rap- 
idly the  friction  surface  p  gains  on  the  area  as  the  diameter 
is  reduced  from  four  feet,  and  also  the  reverse  gain  of  area 
over  friction  surface  as  the  diameter  is  increased  above  four 
feet.  In  pipes  or  circular  closed  channels  flowing  full,  the 
the  same  value  of  d  or  r  always  represents  the  same  length 
of  wet  perimeter,  because  the  parimeter  or  circumference  of 
a  circular  closed  channel  or  pipe  is  always  equal  to  dX3.1416, 
or  to  rX12.5P64.  As  the  area  is  always  as  d2  or  r8  and  as  the 
friction  surface  varies  as  d,  r,  p  or  ^/  a  ,  it  follows  that  the 
same  value  of  d,  r  or  p  in  circular  closed  channels  and  pipes 
flowing  full,  will  always  represent  the  same  value  of  the  area 
and  of  the  wet  perimeter  or  circumference.  Such  circular 
full  channels  may  therefore  be  compared  one  with  another, 
by  simple  proportion,  because  in  such  channels, 


36 


SULLIVAN'S  NEW  HYDRAULICS. 


a  :  a  : :  d1  :  d*,  or  a  :  a  :  :  r*  :  r* 

r:r::d:d,  orr:r::p:p 

p  :p::d  :  d,  or  j/a  :  >/  a::p:p 

In  open  channels,  however,  r  is  not  necessarily  an  index 
of  either  the  extent  of  the  area  or  of  the  perimeter,  and  there- 
fore open  channels  of  different  forms  cannot  be  thus  compared 
one  with  another,  but  in  all  cases  r  expresses  the  ratio  of  a  to 
p  in  the  given  case. 

TABLE  OF  OPEN  CHANNELS. 


Channel              Area,    a 

Peri- 
meter, p 

a 
r=— 
P 

r* 

Flume  10-  x20'  
Mississippi  River... 
Lamer  Canal  
River  Seine  
Chaz  illy  Canal  

200.00 
15911.00 
50.40 

i-o22  00 
11.30 

40.00 
1612.00 
31.00 
518.00 
10.80 

5.00 
9.87 
1.81985 

i-  :^~2 
1.0462 

25.  (C 
97.427 
3.31 

m  n 

1.09453 

While  we  cannot  compare  these  open  channels  one  with 
another  as  in  the  case  of  pipes,  yet  if  we  take  the  data  for  any 
one  pipe  or  for  any  one  open  channel  it  will  be  found  in  any 
case  that 

p  :  r  : :  a  :  r* 

In  other  words  the  friction  e  irface  p  varies  as  r,  and  the  area 
varies  as  r*  in  any  possible  shape  or  form  of  open  or  closed 
channel.  As  p  :  r  : :  a  :  r*  in  any  form  of  channel  it  follows  that 
r  bears  the  same  relation  to  a  and  also  to  p  in  any  given  chan- 
nel or  pipe,  as  it  does  in  any  other  channel  or  pipe,  for  r=  a 

p 

in  any  given  case.  The  properties  which  are  common  to  all 
shapes  and  classes  of  channels  and  pipes  are  that,  in  any 
given  case,  the  area  varies  as  r"  and  the  friction  surface  var- 
ies as  r  or  as  y  a,  regardless  of  the  shape  of  the  pipe  or  chan- 
nel. These  properties  which  are  common  to  all  classes  acd 
forms  of  channels  and  pipes  are  the  only  two  which  affect  the 
coefficients.  Hence  the  form  or  shape  or  size  of  the  pipe  or 
channel  does  not  affect  the  application  of  the  coefficients 
which  vary  with  these  properties  which  are  common  to  all 


SULLIVAN'S  NEW  HYDRAULICS.  37 

possible  forms  of  waterways.  The  tables  were  given  not  only 
to  illustrate  the  above  facts,  but  for  other  reasons  which  will 
be  referred  to  when  the  application  of  certain  formulas  to  open 
channels  is  discussed. 

Brahms  discovered  and  announced  these  common  re- 
lations as  early  as  the  middle  of  the  eighteenth  century, 
but  like  his  successors,  he  mistook  the  total  resistance  for  the 
mean  resistance  or  in  other  words  he  did  not  modify  the 
effects  of  total  resistance  by  that  of  total  acceleration.  Since 
that  time,  coefficients  have  been  made  to  vary  either  as  r  or 
asi/~F~~,  and  also  as  some  other  factor  such  as  slope  or  velocity. 

4.  Coefficients  of  Friction  or  of  Resistance — In  general 
terms  a  coefficient  may  be  defined  as  the  constant  amount  or 
per  cent  by  which  the  head  per  foot  length  of  pipe  or  channel 
must  be  reduced  on  account  of  loss  by  frictional  resistances. 

In  any  constant  diameter,  or  in  any  open  channel  of  con- 
stant hydraulic  radius,  the  friction  will  be  directly  as  the 
total  friction  surface  and  directly  as  the  roughness  of  that 
surface,  and  will  increase  directly  as  the  square  of  the 
velocity.  As  the  amount  of  resistance  per  foot  length  of  pipe 
or  channel  is  always  directly  proportional  to  vs  in  any  given 
case,  it  follows  that  the  amount  of  friction  head  per  foot 
length  (S")  required  to  balance  it  must  also  always  be  direct- 
ly proportional  to  vs — otherwise  one  could  not  balance  the 
other,  and  uniform  flow  could  never  occur.  In  any  given 
pipe  or  channel,  if  the  head  or  slope  increases,  the  square  of 
the  velocity,  and  consequently  the  friction,  will  increase  in 
the  same  ratio,  for  v8  is  always  directly  proportional  to  the 
header  slope.  Hence  the  ratio  of  friction  head  per  foot 
length  S",  to  the  square  of  the  velocity,  v8,  is  necessarily  a 
constant  for  all  heads,  slopes  and  velocities.  The  coefficient 
of  resistance  n,  in  any  given  diameter  or  hydraulic  radius  is 
simply  the  expression  of  this  ratio  of  S"  to  v8.  It  follows 
therefore,  as  this  ratio  is  necessarily  a  constant,  that  a 
change  of  slope  or  velocity  can  have  no  possible  effect  upon 
the  value  of  the  coefficient.  As  long  as  the  diameter  or 
mean  hydraulic  radius  remains  constant,  the  coefficient  of 


210991 


38  SULLIVAN'S  NEW  HYDRAULICS, 

Q  It 

resistance  is  n=— 5-  for  all  slopes  and  velocities,  and   will  be 

constantjfor  all  elopes  and  velocities  because  S"  and  v2  must 
always  vary  exactly  at  the  same  rate.  It  is  evident  that  any 
formula  which  causes  the  coefficient  to  vary  in  any  manner, 
or  to  any  extent  whatever,  with  a  change  of  head,  slope  or 
velocity,  violates  the  law  of  gravity  which  shows  that  H  or  S 
must  be  directly  proportional  to  v2  in  all  cases.  It  also 
violates  the  law  of  friction  which  declares  the  friction  to  be 
always  proportional  to  v8.  The  results  computed  by  such  a 
formula  for  any  given  diameter  with  different  slopes,  or  for 
any  given  open  channel  with  different  elopes  must  necessarily 
be  erroneous  at  least  to  the  extent  that  the  value  of  the 
coefficient  was  made  to  vary  with  changing  slopes.  The 
value  of  the  coefficient  for  any  given  diameter  or  for  any 
given  hydraulic  radius  will  depend  upon  the  degree  of  rough- 
ness of  the  wet  perimeter.  A  rough  perimeter  will  offer 
great  resistance  to  the  flow  and  will  require  a  considerable 
head  or  inclination  to  generate  a  small  velocity.  In  such 

S" 
case  the  ratio  (n)   of  rfwill  be  large,  because  S"  will  be  large 

and  v*  will  be  small.  This  ratio  will,  however,  be  constant 
for  any  given  degree  of  roughness  in  a  pipe  or  channel  where 
r  is  constant. 

5.  Coefficient  of  Velocity— Where  the  discharge  is  free 
in  a  pipe  or  channel,  and  the  value  of  r  remains  constant, 
the  totalhead  will  be  consumed  in  balancing  the  resistance 
and  in  generating  the  velocity  of  flow.  The  resistance  must 
be  balanced  before  flow  can  ensue.  The  resistance  being  as 
va  in  all  cases,  the  coefficients  of  velocity  represents  the  ratio 
of  total  head  H,  to  v8,  or  rather  Sto  va  if  the  discharge  is  free 
If  the  discharge  is  throttled  so  that  a  portion  of  the  head  is 
converted  into  radial  pressure,  then  this  pressure  head  is 
neither  converted  into  velocity  nor  lost  by  resistance.  In  this 

latter  case  the  coefficient  of  velocity  is  the  ratio  oflttpl,  in 


SULLIVAN'S  NEW  HYDRAULICS.  39 

which  Sv  is  the  velocity  head  per  foot  length,  and  S"  is  the 
friction  head  per  foot  length.  Where  the  discharge  is  free, 

S=  total  head  per  foot  length,  and  the  coefficient  of   velocity, 

g 
m—     -.    As    H    or    S    is  always  directly  proportional  to  v8 

(v8— 2gH)  it  follows  that  if  S  be  increased  in  any  given  di- 
ameter or  hydraulic  radius,  v8  will  also  increase  at  the  same 

o 
rate,  and  hence  m=  —-will  necessarily  be  a  constant  in   any 

given  hydraulic  radius  regardless  of  the  value  of  the  slope  or 
velocity.  If  it  be  admitted  that  the  fundamental  laws  of 
gravity,  V8=2  g  H,  and  V=-j/  2g  H  are  correct,  and  that 
the  frictional  resistances  on  any  given  surface  will  increase 
as  va,  it  must  also  be  admitted  that  the  ratio  (m)  of  S  tn  v8  is 
always  constant  after  equilibrium  is  attained,  and  that  as  a 
necessary  result,  the  head,  slope  or  velocity  can  have  no  pos- 
sible effect  upon  the  value  of  the  coefficient  of  velocity. 

6.  The  Law  of  Variation  of  Coefficients—The  coefficient 
of  resistance  n,  and  the  coefficient  of  velocity  m,  have  so  far 
been  considered  only  as  applied  to  a  constant  hydraulic  rad- 
ius or  constant  diameter,  and  it  has  been  shown  that  in  no 
possible  case  can  the  slope  or  velocity  affect  the  value  of  the 
coefficients  in  any  diameter  or  hydraulic  radius.  The  effect  of 
variation  of  hydraulic  radius  or  of  diameter,  upon  the  value 
of  the  coefficient  will  next  be  investigated. 

As  the  area  in  any  given  open  channel  varies  with  r8  and 
in  any  given  circular  closed  channel  or  pipe  as  rs  or  d8,  and 
as  the  wet  perimeter  or  friction  surface  in  any  shape  of  pipe 
or  open  channel  varies  as  r,  d  or  p,  it  follows  that  the  friction 
surface  p  (which  is  as  r  or  d)  must  vary  in  any  given  case  as 
I/!L  Area  is  as  d8  or  r8.  Consequently  •/  a  is  as  d  or  r. 
The  total  resistance  will  be  directly  as  the  total  friction  sur- 
face which  is  as  d,  r  or  ^^7  The  total  area  or  free  flow  in  any 
given  case  varies  with  d8  orr8.  As  area  gains  over  wet  peri- 
meter a  greater  number  of  particles  of  water  are  set  free  from 
the  resistance  of  the  perimeter,  or  acquire  a  head  equal  to  the 


40  SULLIVAN'S  NEW  HYDRAULICS. 

elope  of  the  channel  or  water  surface.  This  does  not  increase 
the  head  of  the  particles  already  free  of  resistance,  but  simply 
adds  to  their  number.  The  number  of  these  unresisted  parti" 
cles  will  increase  directly  as  the  area,  or  as  d8  or  rs.  Total 
acceleration  will  increase  as  the  square  root  of  the  net  free 
head  or  free  flow,  which  is  therefore  as  d  or  r.  Total  retard- 
ation will  be  as  the  square  root  of  the  total  head  lost  by  re- 
sistance. The  head  lost  by  resistance  will  be  directly  as  the 
amount  of  friction  surface,  which  is  as  d  or  r,  but  the  total 
retardation  or  loss  of  acceleration  due  to  this  loss  of  head, 
will  be  as  the  square  root  of  the  head  lost,  or  as  ^"d"  °ri/~rT 
Then  total  acceleration  ie  directly  proportional  to  the 
square  root  of  the  area  or  net  free  head,  or  to  d  or  r,  or  y&. 
Total  retardation  or  loss  of  velocity  is  directly  proportional 
to  the  square  root  of  the  loss  of  head,  or  to  ^  d  or  ^/  r  —  4y  a. 
It  follows  therefore  that  the  mean  loss  of  head,  or  mean  re- 
sistance, of  all'the  particles  of  the  cross-section  taken  as  a 
whole,  will  be  as  the  total  acceleration  modified  by  the  total 
retardation,  or  the  mean  loss  of  head  wi'l  be  inversely  as 
d  /~d~  or  V  T~=  /"r5^  or  i/~&*. 


The  mean  loss  of  head  will  be  inversely  as  \/~r*  or 
because  the  acceleration  increases  as  i/  a  or  d  or  r  while  re- 
tardation is  only  as  -j/  d  or  i/  r,  or  4j/  a.  If  the  value  of 
the  coefficient  of  resistance  n  be  found  for  any  degree  of 
roughness  when  d  or  r=l,  then  it  will  vary  from  this  unit 
point  inversely  as  -/  r8  or  -/  d3,  and  the  general  formula  for 
the  value  of  the  coefficient  becomes,  n=___Xi/  d3,  or  n  = 


These    expressions    are    equivalent   to    n=      r^  r '    or 

n_h"d1/~d" 

Z  v* 
It  is  apparent  that  the  coefficient  of  resistance  or  friction 


SULLIVAN'S  NEW  HYDRAULICS,  41 

n,  will  vary  for  any  given  degree  of  roughness,  only  with  the 
variation  in  the  value  of  -^  cl^or/  ~r«".  For  the  same  de- 
gree of  roughness,  no  other  factor  will  affect  its  value  in  any 
manner. 

Now  as  the  mean  loss  of  head  is  inversely  as  ^/  d8  or 
I/  r8  the  mean  gain  in  head  of  the  entire  cross  section  taken 
as  a  whole,  will  be  directly  as  ^/  d8"  or  -/  T8^  The  mean  veloc- 
ity of  the  entire  cross  section,  or  the  gain  in  the  mean  veloc- 
ity, will  be  directly  proportional  to  the  square  root  of  the  net 
mean  head,  or  to  the  square  root  of  the  gain  in  the  net  mean 


I  y-^ 

head,or-yf  " 

The  friction  head  h"is  therefore  inversely  as  i/r~*~OT  v/lT8" 

The  velocity  head  is  directly  as  v/T1   or  ,/~d». 

The  velocity  of  flow  is  directly  as  f/'d8"  or  f/T8".  It 
is  understood  that  in  these  cases  the  elope  remains  constant. 

The  coefficient  of  velocity,  m,  if  determined  for  any  given 
degree  of  roughness  when  d  or  r  =  1,  will  therefore  vary 
from  this  unit  point  directly  as  v/~r^~  or  i/~d*. 


(9) 


n=J7^=^X'/dl  U0) 

— ^^-^X^ <»> 

— —         ci 

(12) 


_ 

AB  these  coefficients  for  any  given    degree  of   roughness 
vary  only  with  ^"d8   or  v/T^.the  value  of  d   or   r  may  be  in 

Q 

inches  or   feet,  while    the  constant  — g-  may  remain   in   feet 
in  either  case,   or  all  the  values  may  be   expressed  in  metres. 


42  SULLIVAN'S  NEW  HYDRAULICS, 

The  value  of  any  given  constant  slope  is  made  more  effec- 
tive directly  as  v/r«~  increases,  because  the  mean  gain  in 
head  increases  directly  as  ,/ r3  or  the  number  of  unre- 
sisted  particles  increases  as  v/T^T  The  mean  head  and  mean 

velocity  both  gain  without  a  change  in   slope.    This  does  not 

o 

affect  the  constant  ratio — g  because  as  S  is  made  more  effec- 
tive by  an  increase  in  ^/  r3,  v2  increases  in  the  same 
ratio. 

7.     Formula  for  Mean   Velocity— Ry  transposition    in 
equation  (11)  we  have 


VH— x-V/=  V  m  -c18) 

But  for  each  given  degree  of  roughness  of  friction  sur- 
face m  is  a  constant  equal  to  the  ratio-^j,  and  varies  only  as 
l/r8  .  Hence  we  may  take  the  square  root  of  the  reciprocal 


/J_         /     v8 
and  write>-\/  m  =  \8./^i~«  whence 


73 (14) 

In  this  case  C  is  a  constant  which  applies  to  all  pipes  or  open 
channels  of  the  given  degrees  of  roughness  represented  by  C. 

V8 

In  other   words  C   represents  -^-g-which  is  constant  for  any 

given  roughness  and  only  varies  with4;/r3.  If  we  replace  C 
by  K  in  equation  (14)  and  reduce  the  formula  to  the  Chezy 
form,  that  is,  if  we  write 

v=Cv/rsT (15) 

then,  C=K  4i/r  ,  which,  when  multiplied  by  y^rs  equals 
C  VrT~1/S7~In  the  Chezy  form  it  is  seen  that  C=K  Vr~I 
or  in  other  words,  that  for  any  given  degree  of  roughness  of 
friction  surface  represented  by  K,  the  coefficient  C  will  vary 
only  as  4^/r  . 


SULLIVAN'S  NEW  HYDRAULICS.  43 

If  we  write  simply.  v=CVS,  then  C=K41/r«. 

The  result  is  the  same  in  either  case. 

The  value  of  C  in  any  formula  in  the  Chezy  form  must 
yary  only  as  the  roughness  and  as  J/r  or  $/d. 

If  a  series  of  pipes  or  open  channels  of  equal  roughness 
be  selected,  it  will  be  found  that  C:  C: :  £/r~:  e/r~regardless 
of  the  slopes  or  dimensions  of  the  channels.  If  C  fails  to  vary 
only  as  J/r  in  the  Chezy  form  of  formula  for  a  series  of  chan- 
nels of  equal  roughness,  then  it  will  be  found  that  C  y/rs  will 
not  equal  v.  This  will  be  illustrated  by  the  following  pairs 
of  open  channels — each  pair  being  nearly  equal  in  roughness, 
but  varying  in  the  values  of  the  slope  and  hydraulic  radius. 

8.  Variation  of  C  Illustrated— In  the  following  tables 
we  shall  give  the  values  of  our  C  as  found  by  the  formula 

C=-»l =>  for  each  channel.    We  will  also  give  the   value 

of  the  Chezy  C  for  each  channel  which  is  required  to  make 
C  v/rs=v.  It  will  be  found  in  each  case  that  barring  the 
slight  difference  in  the  degree  of  roughness  in  each  pair  of 
channels  the  Chezy  C  which  will  cause  C  ^/rs  to  equal  v, 
will  vary  only  as  J/r  ,  and  their  values  in  such  class  of  chan- 
nels may  be  found  or  compared  by  the  simple  proportion 
C  :  C  ::  t/F~:  J/rT  The  values  of  the  Chezy  C  were  deter- 
mined by  the  formula,  C= — —  in  all  the  following  tables' 

V  rs 

The  values  as  given  in  the  translation  of  Ganguillet  and  Kut- 
ter  are  only  approximate,  being  sometimes  in  error  by  as 
much  as  12  or  15.  The  values  of  Kutter's  n,  or  constant  co- 
efficient of  roughness,  are  also  given  for  each  channel.  These 
values  of  n  are  transcribed  from  Hering  &  Trautwines  Trans- 
lation, second  edition.  These  values  of  n  show  that  it  is  not 
a  constant,  but  is  an  auxiliary  quantity  which  must  be  used 
as  r  and  s  vary  in  order  to  balance  the  erroneous  variation  of 
C  with  the  slope. 


SULLIVAN'S  NEW  HYDRAULICS. 


Pair  No.  1.     Slopes  equal.     Radii    vary    slightly.     R>  1 
meter. 


NAME  OF 
CHANNEL 

R 

FEET 

f/R~ 

FEET 

S 
SLOPE 

V 
FEET 
SEC. 

SULLI- 
VAN'S 

c 

KUT- 

TER'S 
n 

CHEZY 
C 

Seine 
(PariB) 
Seine 
(Paris) 

9.50 
10.90 

1.7556 
1.8170 

.00014 

.00014 

3.37 
3.741 

52.85 
52.71 

.0240 
.0238 

92.40 
95.77 

The  channel  where  R=10.9G  is  very  slightly  rougher,  as 
sbown  by  Sullivan's  C,  than  the  firet  channel.  *As  the  slopes 
are  equal,  Kutter's  n  has  to  vary  only  with  the  slight  differ- 
ence in  values  of  r,J£  order  that  c  ^/rs"  will  equal  v,  and  in 
order  that  c:  c: :  £/~r~:  $/r~. 

Pair  No.  2.  Slopes  nearly  equal.  Small  difference  in  R. 
R  >  1  meter. 


NAME  OF 
CHANNEL 

R 

FEET 

t/R 

FEET 

S 

SLOPE 

V 

FEET 
SEC. 

SULLI 

VAN'S 
C 

KUT- 
TER'S 
n 

CHEZY 

C 

Seine 
(Triel) 
Seine 
(Poissy) 

12.40 
15.90 

1.876 
2.000 

.00060 
.00062 

2.359 
2.911 

46.12 
46.43 

.0295 
.0285 

86.43 
92.70 

The  Chezy  Ovaries  only  as  £/r  in  Pair  2.  It  is  not  af- 
fected by  the  slight  difference  in  elope.  Where  R=15.90  the 
channel  is  very  slightly  smoother  than  where  R=12.40.  Yet 
Kutter's  n  must  be  reduced  because  of  the  slight  increase  of 
slope  and  hydraulic  radius.  As  Kutter's  C  will  increase  with 
decrease  in  elope  where  R  is  greater  than  1  meter,  or  3281 
feet,  n  must  be  increased  where  R=12.40  because  this  slope  is 
least,  and  if  n  were  taken  as  a  constant  for  both  channels,  it 
would  make  C  too  great  for  the  first  channel. 

Pair  No  3. 
equal  nearly. 


R  >  1  meter.    Rand   S   vary.      Roughness 


NAME  OF 
CHANNEL 

R 

FEET 

FEET 

S 
SLOPE 

V 
FEET 
SEC. 

SULLI- 
VAN'S 
C 

KUT- 
TER'S 

n 

CHEZY 
C 

La 
Fourche 

15.70 

1988 

.0000438 

2.798 

53.40 

.0205 

106.30 

Mississip- 
pi Rmr 

72.00 

2.913 

.0000205 

5.929 

53.00 

.0277 

154.30 

SULLIVAN'S  NEW  HYDRAULICS.  45 

Note  the  difference  in  value  of  Kutter'e  n  for  these  two 
channels  of  equal  roughness. 

In  Pair  No.  3  R  is  greater  than  one  meter,  and  in  this 
case  Kutter's  C  will  increase  as  slope  decreases.  As  the 
slope  of  the  Mississippi  river  is  much  less  than  that  of 
Bayou  La  Fourche,  if  n  were  used  as  a  constant  for  both,  the 
value  of  Kutter's  C  would-  be  greatly  too  large  for  the  Mis- 
sissippi. Therefore,  in  order  to  balance  the  error  of  increase 
in  Kutter's  C  with  decrease  in  slope,  the  value  of  n  must  be 
increased  in  proportion  as  slope  decreases.  Otherwise  his 
GI/  rs  will  not  equal  v.  It  is  seen  from  Pair  No.  3  that  when 
the  required  value  of  the  Chezy  or  Kutter  C  is  obtained 
which  will  make  Cv/~re~=\,  then  C:C:  :  V~T  :  V~,  re- 
gardless of  the  difference  in  slope.  Kutter  admits,  in  his 
work  on  Hydraulics  (pages  99  and  132)  that  n  is  not  a  con- 
stant for  the  same  degree  of  roughness  if  there  is  much  vari- 
ation in  the  dimensions  of  the  channels  to  which  it  is  applied. 
His  n  might  be  made  a  constant  like  our  C  for  each  degree 
of  roughness,  and  regardless  of  the  dimensions  of  the  chan- 
nels, if  it  were  made  to  vary  only  as  J/TT,  for  all  slopes  and 
all  dimensions  of  channels,  whether  R  were  greater  or  less 
than  one  meter.  It  is  absurd  that  C,  and  consequently  the 
velocity,  should  be  proportionately  less  for  a  steep  slope  in  a 
large  channel  than  for  a  small  slope.  Of  course  the  value  of 
I/  S  remains  in  any  case,  but  decrease  in  C  as  S  increases  in 
large  channels  amounts  to  reducing  the  actual  value  of  S  by 
the  amount  that  C  is  there  made  to  decrease.  It  cannot  be 
justified  upon  any  sound  theory,  and  the  above  tables  show 
that  it  is  not  sustained  by  fact.  It  is  equally  erroneous  that 
C  will  increase  with  an  increase  in  slope  in  small  chan- 
nels where  R  is  less  than  one  meter,  and  in  which  the 
ratio  of  friction  surface  to  the  quantity  of  water  passed 
is  much  greater  than  in  large  channels.  The  laws  of  grav- 
ity and  of  friction  do  not  reverse  themselves  at  the  point 
where  R=l  meter,  nor  at  any  other  value  of  R.  As  Kut- 
ter's n  is  not  a  constant  for  the  same  degree  of  roughness 
where  the  slopes  vary  or  where  R  varies,  it  is  very  mislead- 


SULLIVAN'S  NEW  HYDRAULICS. 


ing  when   viewed  as   an    index    of    roughness,  which  is  sup- 
posed to  be  its  special  function. 
Pair  No.  4.  R<   1  Meter.    Roughness  Equal.    R  and  S  Vary 


Name  of 

R 

VK~ 

S 

v 

Sulli 

Kut- 

Chezy 

Channel 

Feet 

Feet 

Slope 

Ft  Sec 

van's  C 

ter's  n 

C 

Rhine  Forest 

0.42 

.8051 

.0142 

2.332 

37.50 

.0337 

30.13 

Simme 

Canal 

1.32 

1.072 

.0170 

5.993 

37.37 

.0361 

40.06 

In  Pair  No.  4,  R  is  lesa  than  one  meter  in  either  channel. 
For  this  reason  Kutter's  C  will  increase  with  increase  of 
slope.  Hence  the  steeper  the  slope  becomes  where  R  is  less 
than  one  meter,  the  greater  we  must  increase  the  value  of 
his  n  in  order  to  cut  down  this  unnatural  increase  in  C.  We 
find  by  simple  proportion  in  Pair  No.  4.  as  in  all  other  cases 
where  the  roughness  is  equal,  that  C:C:  :  *|/r:*-l/r,  simply, 
and  regardless  of  difference  in  slope.  Kutter's  n  must  be 
trimmed  or  increased  in  such  manner  as  to  cause  C  to  vary 
only  as  *y'r,  otherwise  his  C^/rs  will  not  equal  v.  It  is  there- 
fore neither  a  constant  nor  an  index  of  roughness,  but  is  an 
uncertain  and  misleading  quantity.  See  Kutter'e  discussion 
of  the  variation  of  his  n  at  pages  99,  110  and  132  of  Hering 
and  Trautwine's  edition  of  Kutter'e  work.  Also  see  Trans- 
lators preface. 
Pair  No.  5.  R  and  S  vary.  Roughness  Equal. 


Name  of 
Channel 

R 

Feet 

t/r 
Feet 

S 
Slope 

V 

Ft  Sec 

Sulli- 
van's C 

Kut- 
ter's r 

Chezy 
C 

Grosbois  Ca- 
nal 
Seine  (Paris) 

1.71 
14.50 

1.143 
1.951 

.000441 
.00014 

1.51 
4.232 

48.08 
48.12 

.0284 
.0255 

55.50 
93.92 

In  Pair  No.  5,  the  value  of  R  is  less  than  1  meter  in  one 
case,  and  greater  in  the  other,  and  there  is  a  difference  in 
slope  also.  Notwithstanding  both  these  facts,  C  must  vary 
only  as  4|/r  as  shown  in  the  table,  or  C^/rs  will  not  equal  v. 

Sullivan's    C  in    all  the  above  tables  is  C=A/-; — =^;  and  ap- 


SULLIVAN'S  NEW  HYDRAULICS.  47 

plies  in  the  formula,  v=C$/r*~  ^ST"  Its  unit  value  is  con- 
stant for  all  slopes  and  all  dimensions  of  pipes  or  open 
channels  of  the  same  degree  of  roughness.  It  is 
simply  the  square  root  of  the  reciprocal  of  m.  It 
has  been  shown  that  slope  or  velocity  cannot  affect  the  value 

Q 

of  m,  as  it  is  the  expression  of  the  ratio-^j  •  ItB  numer- 
ical value  depends  only  upon  the  degree  of  roughness  of  peri- 
meter. The  formula  for  m  or  n  or  C  as  heretofore  given,  will 
give  the  unit  value  of  the  coefficient  directly,  that  is,  its  value 
for  r  or  d=l.  It  therefore  does  not  matter  whether  the  for- 
mula for  ascertaining  the  coefficient  is  applied  to  the  data  of 
a  very  small  or  very  large  channel, the  result  will  be  the  value 
of  the  coefficient  for  r=l,  or  d=l,  as  the  case  may  be.  From 
this  unit  point  the  coefficient  varies  with  the  inverse  value 
of  i/  r8  or  y'  d8  if  it  is  n  that  is  sought.  The  coefficient  m 
of  velocity,  varies  from  the  unit  value  as  found  by  formula 

form,  directly  as  i/~d*or  l/~r^.  The  variation  of  C  will  be 
as  the  {/T~if  the  formula  is  written  v=C1/~r6,  or  if  it 
written  v=C  X  V~  i/~rs.  If  we  write  v=C  ,/S"  then  C 
must  vary  as  f/r3.  This  latter  form  is  equivalent  to  the 
form  v=C  X  V~**  »/~S. in  which  C  is  the  constant  for  any 
given  degree  of  roughness  of  perimeter.  This  last  form  has 
been  adopted  in  all  the  foregoing  and  following  tables.  For 
the  reason  that  m  or  C,  as  found  by  formula  from  the  data 
of  guagings  will  be  the  unit  value,  and  will  differ  in  value 
only  as  the  degree  of  roughness  differs,  the  mere  develop- 
ment of  the  unit  values  of  the  coefficient  for  a  series  of  pipes 
or  open  channels  will  at  once  classify  such  pipes  or  chan- 
nels, and  exhibit  their  relative  degrees  of  roughness.  Those 
which  give  like  values  of  the  coefficient  are  of  similar  degrees 
of  roughness,  because  the  unit  value  of  the  coefficient  is  not 
affected  by  any  element  or  factor  except  the  degree  of  rough- 
ness. 

The  coefficient  C  or  m  does  not,  and  should  not,  vary,  ex- 


48  SULLIVAN'S  NEW  HYDRAULICS. 

cept  as  the  roughness  of  perimeter  varies.  For  this  reason 
our  in  or  G  is  an  absolute  index  of  the  roughness  for  it  cannot 
vary  with  any  other  factor.  We  have  shown  that  the  effect- 
ive value  of  the  slope  S  is  increased  as  -^  r3  increases,  because 
the  net  meau  head,  or  net  gain  in  area  over  friction  surface 
is  as  >/  r3  .  But  whatever  increases  or  makes  the  mean  head, 
or  S,  more  effective,  must  alao  increase  the  value  of  v*  in  the 
same  ratio. 

The  effective  slope  S,  is  as  S  i/T5"  ,  and  the  mean    veloc- 

ity is  as,.  /S  i/  r3  .  Now  in  the  formula  for  m  or  C.  m=  -  ^  — 
and  C=  |  —  Y  .  In  either  formula  an  increase  in  the  value 


=    |  —  Y 

^S/r3 


r3  will  cause  the  value  of  v8  to  increase  in  the  same 
ratio.  It  is  then  apparent  that  where  the  values  of  Sv/T3"  are 
equal,  the  velocities  must  be  equal  unless  the  resistances 
caused  by  roughness  of  perimeter  are  greater  in  the  one  case 
than  in  the  other.  It  is  also  apparent  from  an  inspection  of 
the  formula  for  m  or  G  that  as  vs  will  increase  in  the  same 
ratio  as  Sv/~r*"  increases,  m  or  C  will  be  constant  for  all  val- 
ues of  r  or  d  if  the  roughness  of  perimeters  is  the  same. 

In  the  velocity  formula,  v=CV~rTXv/~ST  we  see  that  the 
mean  velocity  increases  not  only  as  ,/  S  but  also  as  the 
square  root  ot  ^/  r3  ,  which  is  $/  r3  ,  not  because  m  or  C  var- 
ies, but  because  the  value  of  S  is  made  more  effective  as  v/~r^ 
increases. 

9.    Practical  Determination  of  Coefficients  of  Resistance. 

The  resistance  to  flow,  or  loss  of  head  by  friction,  is  exactly 
equal  to  the  amount  of  head,  pressure,  or  force  required  to 
balance  it.  In  a  pipe  of  uniform  diameter  and  roughness  the 
friction  will  be  the  same  in  one  foot  length  of  pipe  as  in  any 
other  foot  length,  hence  the  total  friction  will  be  directly  as 
the  length  and  roughness  of  the  pipe.  Friction  in  any  given 
diameter  and  roughness  of  pipe  will  increase  with  the  square 


SULLIVAN'S  NEW  HYDRAULICS.  49 

of  the  velocity.  Hence  the  head  lost  by  friction,  or  the  head 
which  is  consumed  in  balancing  friction,  must  also  increase 
as  the  square  of  the  velocity.  The  friction  or  loss  of  head  for 
any  given  velocity  in  different  diameters  will  be  inversely  as 
y'  d8  or  v/r8,  because  total  acceleration  is  proportional  to 
the  square  root  of  the  area,  or  to  d  or  r,  while  total  retarda- 
tion is  proportional  only  to  j/cf  or  i/F".  Hence  the  mean  loss 
of  head  of  all  the  particles  of  water  will  be  inversely  propor- 
tional to  the  resultant  of  total  acceleration  and  total  retarda- 
tion, or  to  dy/  d  =  i/d8,  or  ri/r  =  i/r8-  (See  columns 
headed  d.y/d,  and  "Relation  of  P  to  A,"  in  table  of  circles, 
ante,  §3). 

The  mean  of  many  experiments  shows  that  a  cast  iron 
pipe  of  ordinary  density  or  specific  gravity,  one  foot  in  diam- 
eter and  clean,  will  require  a  total  head  of  one  foot  in  a  length 
of  2,500  feet,  in  order  to  cause  it  to  generate  a  velocity  of  one 
foot  per  second.  The  discharge  being  free,  it  is  evident  that 
the  total  head  of  one  foot  has  been  lost  by  resistance  except 
that  part  of  the  one  foot  head  which  remained  to  generate 
the  mean  velocity  of  one  foot  per  second.  As  the  velocity 
head  is  not  lost  by  resistance,  and  as  we  wish  to  determine 
the  numerical  value  of  the  coefficient  of  resistance  n,  the  ve- 
locity head  must  be  deducted  from  the  total  head  of  one  foot 
in  order  to  find  the  total  head  lost  by  friction.  By  the  law  of 
gravity  we  find  that  the  head  which  generates  any  given  ve- 


In  the  case   we  are   now  considering   vff=l,  and  conse- 

100 
quentlv  the  velocity  head  hv=-gj^=.01552795  feet.    Deduct- 

ing this  velocity  head,  which  was  not  lost,  from  the  total 
head  of  one  foot,  and  we  find  that  the  total  head  lost  by  fric- 
tion in  the  2,500  feet  of  12-inch  pipe  while  v*=l  was  equal  to 
1.00—  .01552795=.98447205  feet.  Therefore  the  head  lost  per 

.98447205 
foot  length  of  pipe   while  vj=l,    and    d=l,  was  •  —  2500  — 


753 


50  SULLIVAN'S  NEW  HYDRAULICS. 

=.00039379  feet=n.  As  the  friction  will  be  as  the  number 
of  feet  length  of  the  constant  diameter,  and  will  increase  as 
VT,  then,  as  long  as  d  remains  constant,  the  total  head  in  feet 
lost  by  friction,  h"=n  X  I  Xv8.  But  if  the  value  of  d 
changes,  or  the  formula  is  to  be  applied  to  a  pipe  of  like 
roughness,  but  of  a  different  diameter,  we  have  seen  that 
the  friction  will  be  inversely  as  ^/  d3.  Henee  the  general 
formula  which  will  apply  equally  to  all  diameters  of  this 
given  degree  of  roughness  will  be 

n  *  y8 


v/d3  •)/  d8 

We  might  have  found  the  value  of  n  directly  by  applying 
formula  (10)  (§  6). 

—  Xv/^F=mX-9845 (10) 

v2 

Q // 

As  the  ratio  ot—j  is  always  constant  for  any  given  de- 
gree of  roughness,  regardless  of  slope  or  velocity,  and  as  it 
varies  from  the  unit  point,  or  d=l,  and  v*=l,  only 
as  v/cl1  varies,  we  may  find  the  unit  value  of  the  co- 
efficient from  any  diameter  and  velocity  whatever. 

S" 
It   is    simply    necessary     to    find    the   ratio-^  in     any 

case,  and  when  the  value  of-|j  is  multiplied  byl/~df ,  the  re- 
sult will  be  the  unit  value  of  n.  When  this  unit  value  of  n 
is  inserted  in  formula  (16)  it  is  made  to  vary  inversely  as  ^/  d* 
as  exhibited  in  formula  (16).  To  make  it  appear  more  clearly 
we  write 

n 


It  consequently  does  not  matter  what  head,  diameter  or 
velocity  we  may  select  for  the  purpose    of    finding  the  unit 


SULLIVAN'S  NEW  HYDRAULICS.  51 

value  of  n.  The  formula  for  n  will  always  give  the  unit 
value,  regardless  of  the  size  of  the  pipe  to  which  the  for- 
mula is  applied.  As  the  unit  value  of  n  is  not  affected  by 
any  factor  except  the  degree  of  roughness,  it  is  a  faithful  in- 
dex of  roughness,  and  when  the  value  of  n  for  a  series  of  dif- 
ferent classes  of  perimeter  has  been  found,  it  exhibits  the 
direct  difference  in  roughness  per  unit  of  perimeter,  between 
the  different  classes. 

10.  Conversion  of  the  Coefficient.— The  coefficient 
may  be  determined  in  terms  of  diameter  in  feet,  or  diameter 
in  inches,  or  in  terms  of  r  instead  of  d,  or  in  terms  of  cubic 
feet  or  gallons.  If  the  value  of  n  has  been  found  for  any 
given  degree  of  roughness,  it  may  be  converted  to  any  de- 
sired terms.  Thus,  if  the  value  of  n  has  been  found  in  terms 
of  d  in  feet,  as  above,  it  may,  be  converted  to  terms  of  r  in 
feet  by  simply  multiplying  it  by  0.125  or  dividing  by  eight. 
If  n  was  originally  found  in  terms  of  r,  and  it  is  desired  to 
convert  it  to  terms  of  d  in  feet,  multiply  by  eight.  If  n  is  in 
terms  of  d  in  feet,  it  may  be  converted  to  terms  of  d  in  inches 
by  multiplying  by  v/(r2)ff=41.5692.  ^s  n,  for  any  given  de- 
gree of  roughness,  varies  only  with  ^/"d^  the  value  of  d  may 
be  in  meters,  inches  or  feet,  as  may  be  most  convenient,  h", 
I  and  v*  may  remain  in  feet  or  meters. 

m=-;and  n  =  mX-9845,  for  any  given  degree  of  rough- 


//.  Determination  of  Coefficients  of  Velocity.— We  have 
just  seen  that  a  coefficient  of  resistance  (n)  represents  only 
the  head  per  foot  length  of  pipe  which  is  lost  or  consumed  in 
balancing  the  resistance  to  flow.  A  coefficient  of  velocity, 
however,  must  represent  not  only  the  head  per  foot  length 
required  to  balance  the  resistance,  but  also  the  head  per 
foot  length  required  to  generate  the  velocity  of  flow,  or  it 
must  represent  S"-|-Sv  in  any  case.  If  the  diameter  or 
hydraulic  radius  is  constant,  and  the  discharge  is  free  and 
full  bore,  the  total  head  per  foot  length  S,  will  be  converted 


52  SULLIVAN'S  NEW  HYDRAULICS. 

into  velocity  of  flow  except  that  part  of  S  which  is  consumed 
in  balancing  friction.  In  this  case,  S"+Sv=S,  and  S  must 
be  used  in  the  formula  for  determining  the  value  of  m  —  the 
coefficient  of  velocity.  Where  the  discharge  is  partially 
throttled,  as  by  a  reducer  at  discharge,  or  by  a  valve  partly 
closed,  only  a  part  of  the  total  head  per  foot  length  will  be 
consumed  by  resistances  and  in  generating  velocity,  and  the 
remainder  of  the  head  will  remain  as  radial  pressure  within 
the  pipe.  As  the  head  due  to  this  pressure  is  neither  lost 
by  resistance  nor  engaged  in  generating  velocity  of  flow,  it 
has  no  connection  with  the  value  of  the  coefficient  of  velocity 
m.  If  the  discharge  is  free,  then 

H  v/r«         S  S,/r«  n 

m=—  ^s—  ^iXv/r«=    -^I-=;9845  —  <17> 
If  the  discharge  is  throttled,  then 


For  the  ordinary  cast  iron  pipe  described  in  section  9, 
the  coefficient  of  velocity  would  be 

-WM.m  terms  of  d  in  feet. 

The  coefficient  m  may  be  converted  to  terms  of  d  in 
inches,  or  r  in  feet  or  to  any  other  terms  in  the  same  manner, 
and  by  the  use  of  the  same  multipliers,  as  n  may  be  con- 
verted. (See  §  10) 

The  velocity  coefficient  m  applies  to  open  or  closed  chan- 
nels alike  and  its  unit  value  depends  only  on  the  degree  of 
roughness  of  perimeter.  The  value  of  m  as  found  by  the  form- 
ula is  always  the  unit  value,  and  is  equally  as  accurate  an 
index  of  roughness  as  is  the  coefficient  n.  The  remarks  in 
regard  to  n  in  this  respect  (§  9)apply  to  m  with  equal  force. 

The  coefficient  m  is  to  be  used  in  the  formula, 


4 


SULLIVAN'S  NEW  HYDRAULICS.  53 


If  m  was  determined  in  terms  of  r,  it  must  not  be  used 
in  formula  (20)  which  is  in  terms  of  d,  until  it  has  been  con- 
verted to  like  terms  with  those  in  the  formula.  If  m  is  in 
terms  of  d  in  inches,  then  d  in  the  formula  must  also  be  in 
inches.  In  other  words  m  must  be  in  the  bame  terms  as  the 
formula  in  which  it  is  used  is  expressed. 

The  value  of  m  in  terms  of  d  in  feet  for  average  cast  iron 
pipe  is  m=.0004.  If  it  is  desired  to  use  C  instead  of  m  then 

=  50.00    and 


The  value  of  C  may  be  found  directly  and  without  refer- 
ence to  m  by  the  formula 


0=4 


or  C  = 


'8,/r* 

This  will  give  the  unit  value  of  C  directly,  and  C  is  a 
constant  like  m  or  n,  which  depends  on  the  roughness  of 
perimeter. 

If  we  have  m=.0004  for  ordinary  cast  iron  pipe,  in  terms 
of  diameter  in  feet,  we  may  convert  it  to  terms  of  r  in  feet  by 

0004 
simply  dividing  by  8.     We  then  have  -^ — =  .00005  =  m    in 

terms  of  r  in  feet.  We  may  convert  m  to  C  in  terms  of  r  in 
feet  by  taking  the  square  root  of  its  reciprocal  in  terms  of  r, 
and  we  have 


v/20000  =  141.42  =  C  in  terms  of  r. 

Then,    v  =  C  {/r^  ,/S. 

The  unit  values  of  n,  m  and  C  may  be  found  in  all  classes 
of  pipes  and  channels,  and  may  be  converted  at  pleasure  as 
shown.  The  law  governing  the  flow  of  water  and  the  value 
and  variation  of  the  coefficients,  is  exactly  the  same  in  open 


54  SULLIVAN'S  NEW  HYDRAULICS, 

channels  as  in  pipes.  The  same  formulas  apply  to  all  equally 
well  BO  far  as  the  coefficients  and  the  formulas  for  flow  are 
concerned.  Of  course  the  unit  value  of  the  coefficient  must 
be  found  experimentally  for  each  class  or  degree  of  Toughness 
of  friction  surface.  When  the  unit  value  of  the  coefficient  is 
determined  for  any  given  degree  of  roughness,  it  then  applies 
to  all  forms  and  dimensions  of  pipes  and  channels  which  fall 
within  that  degree  of  roughness.  These  remarks  apply  to  n, 
m  and  C  alike.  The  roughness  or  smoothness  of  perimeter 
affects  the  flow  in  a  large  river  in  the  same  manner  as  in  a 
email  canal.  In  a  large,  deep  river  the  area  of  the  cross- 
section  of  the  column  of  water  is  greater  in  proportion  to  the 
wet  perimeter  than  in  a  small  stream,  and  hence  the  ratio  of 
free  particles  of  water  is  greater  than  in  small  channels,  but 
the  effect  of  roughness  of  perimeter  is  the  same  in  both  cases. 
The  unit  value  of  m  and  C  distinctly  establish  these  facts. 
It  is  the  influence  of  the  great  values  of  r  in  large  rivers  that 
has  led  some  hydraulicians  to  conclude  that  the  character  of 
the  perimeter  does  not  materially  affect  the  flow  in  such 
streams. 

12.— Coefficients  Affected  by  Specific  Gravity,  or  Den- 
sity of  Material.— In  a  series  of  experiments  with  new,  clean 
cast  iron  pipes  the  writer  was  perplexed  by  the  fact  that  one 
12  inch  new,  clean  pipe  would  not  generate  the  same  mean 
velocity  as  another  new,  clean  12  inch  pipe,  when  the  con- 
ditions were  exactly  the  same  in  each  case.  The  difference 
was  so  great  in  the  case  of  one  pair  of  12  inch  new  pipes, 
that  the  experiment  was  repeated  a  number  of  times,  but 
always  with  the  same  result.  As  no  other  explanation  could 
be  given  the  writer  concluded  to  ascertain  if  it  was  caused  by 
the  difference  in  density  or  specific  gravity  of  the  two  pipes, 
which  were  from  different  foundries.  The  shells  were  of 
equal  thickness,  but  on  weighing  a  few  lengths  of  the  pipe 
from  each  lot,  it  was  found  that  the  pipe  which  generated 
the  least  velocity  was  much  lighter  than  the  other.  The 
investigation  thus  begun  led  to  experiments  with  pipes  of 
different  metals  and  different  specific  gravities.  The  results 


SULLIVAN'S  NEW  HYDRAULICS,  55 

then  obtained  seem  to  confirm  the  correctness  of  the  view 
that  the  density  of  the  friction  surface  has  a  marked  influ- 
ence upon  the  flow  and  upon  the  value  of  the  coefficient. 
There  may  be  some  difference  also  between  the  values  of  the 
coefficient  for  a  surface  of  granular  metal  and  a  surface  cf 
fibrous  metal,  although  the  specific  gravities  of  the  two 
metals  may  be  equal.  It  appears  that  the  flow  over  earthen 
perimeters  of  equal  regularity  of  cross-section  will  be  affected 
by  the  nature  and  specific  gravity  of  the  particular  kind  of 
earth.  The  flow  in  a  cement  lined  pipe  or  channel  which  is 
clean  and  free  of  fine  silt,  will  be  affected  by  the  fineness  of  the 
cement  and  also  of  the  sand  used,  as  well  as  by  the  propor- 
tion of  sand  to  cement  in  the  mortar  lining.  Even  in  pure 
cement  linings,  it  is  noticed  that  the  flow  will  be  affected  by 
the  quality  and  fineness  of  the  cement  used.  Classification  of 
perimeters  is  therefore  difficult. 

It  is  stated  by  Professor  Merriman  that  "it  is  proved  by 
actual  gaugings  that  a  pipe  10,000  feet  long  and  one  foot  in 
diameter  discharges  about  4.25  cubic  feet  per  second  under 
a  head  of  100  feet.  The  mean  velocity  then  is 

v=—  —  -=5.41  feet  per   second."      ("Treatise  on   hy- 

draulics." page  165,  fifth  edition.)  It  will  be  noted  that  the 
character  of  the  pipe,  whether  cast  iron,  wrought  iron,  riveted 
or  welded,  coated  or  uncoated,  is  not  mentioned.  It  was 
certainly  a  remarkably  smooth  pipe.  If  the  value  of  the  co- 
efficient m  is  developed  for  this  pipe  we  shall  have 


m=      -Xv/  d3=.00034165,  in  terms  of  d   in  feet. 

m=  ^-Xi/  r«=.00004270625,  in  terms  of  r  in   feet. 

The  average  value  of  m  for  clean  cast  iron  pipe  is 

m=.01662768,  in  term  of  d  in  inches. 

m=.000i  in  terms  of  d  in  feet. 

m=.00005  in  terms  of  r  in  feet. 
The  writer  made  a  numberof  experiments   with  6",    12' 


56  SULLIVAN'S  NEW  HYDRAULICS. 

and  24"  cast  iron  pipes  which  were  new  and  absolutely  clean 
and  of    the    greatest  density    that  the   writer    has  ever  dis- 
covered before  or  since  in  cast    iron   pipes.      The   water   was 
pure  mountain   water  from  the  melting  snow  on  the  granite 
hills.     The  pipes  were    laid   straight   and   perfectly   jointed, 
and  the  discharge  was  perfectly  free,  into  a   large   measuring 
tank.    Under  these  perfect  experimental  conditions,  the  value 
of  m  as  developed  by  the  three  pipes  was 
m=. 000368  in  terms  of  d  in  feet. 
m=.000046  in  terms  of  r  in  feet. 

Such  favorable  conditions  as  these  scarcely  ever  occur  in 
actual  water  works  building,  and  do  not  continue  if  they 
originally  exist. 

In  later  experiments  with  new  clean  cast  iron  pipes  of  in- 
ferior quality  and  very  low  specific  gravity,  the  values  of  the 
coefficient  of  flow  developed  were 

m=.01721  in  terms  of  d  in  inches:— C=7.622. 

m=.000414  in  terms  of  d  in  feet:— C=49.14. 

New,  clean  cast  iron  pipe  of  average  weight  per  cubic 
unit  as  long  as  it  remains  clean  gives, 

m=.01663  in  terms  of  d  in  inches:— C=7 .755. 

m=.0004  in  terms  of  d  in  feet:— C=50.00. 

m=.00005  in  terms  of  r  in  feet:— C=141.42. 

It  is  therefore  evident  that  where  the  pipes  are  made  of 
the  same  class  of  metal  and  are  new  and  clean,  the  value  of 
the  coefficient  will  bear  a  close  relation  to  the  specific  grav- 
ity, or  density,  of  the  pipe  metal.  The  fact  that  clean  leaden 
or  brass  pipe  will  generate  a  much  greater  velocity  of  flow 
under  the  same  conditions  than  will  a  clean  iron  pipe  of 
equal  diameter  can  be  accounted  for  in  no  othei  manner  than 
the  difference  in  specific  gravity  of  the  different  metals. 

These  facts  demonstrate  the  important  influence  of  even 
very  small  degrees  of  roughness  of  perimeter  upon  the  flow 
and  consequently  upon  the  value  of  the  coefficients.  Low 
specific  gravity  in  metal  indicates  that  it  is  porous  and  its 
surface  is  affected  by  innumerable  small  cavities,  rendering  it 


SULLIVAN'S  NEW  HYDRAULICS.  57 

irregular.  The  specific  gravity  of  cast  iron  varies  from  6.90 
to  7.50;  of  steel,  from  7.70  to  7.90;  of  wrought  iron  from  7.60 
to  7.90. 

While  the  specific  gravity  of  a  metal,  or  of  stone  or  brick, 
01-  earth  where  the  cross  section  is  equally  uniform,  undoubt- 
edly affects  the  flow,  yet  other  substances  of  much  less  spec- 
ific gravity,  when  applied  as  a  lining  or  coating,  will  greatly 
increase  the  flow.  Thus  the  specific  gravity  of  asphaltum 
varies  from  1  to  1.80  according  to  its  purity,  and  an  asphaltum 
coated  pipe  will  generate  a  much  higher  velocity  of  flow  than 
a  clean  iron  pipe.  The  coefficients  developed  by  asphaltum 
coated  pipes,  however,  vary  like  cement  lined  pipes,  with  the 
quality  of  the  material,  or  the  proportion  of  pure  asphaltum 
to  the  other  ingredients  u^ed  in  the  manufacture  of  the  coat- 
ing comtound.  It  would  appear  therefore  that  while  the 
specific  gravity  of  one  metal  may  be  compared  with  that  of 
another  metal,  or  the  specific  gravity  of  one  class  of  as- 
phaltum coating  compound  may  be  compared  with  another, 
as  to  its  probable  resistance  to  flow,  we  cannot  compare  ma- 
terials of  wholly  different  natures  with  each  other,  and  judge 
of  the  relative  resistance  by  the  respective  densities.  The 
values  of  m  for  asphaltum  coated  double  riveted  wrought  iron 
pipe  when  new  varies  with  quality  of  the  coating  as  follows: 
m=.000036  in  terms  of  r  in  feet,  to  m=.000044. 
m=.000288  in  terms  of  d  in  feet,  to  m=. 000352. 

The  average  value  of  m  for  such  coating  while  in  prime 
condition  may  be  taken  as  m=.00033,  in  terms  of  d  in  feet. 
The  average  value  of  the  coefficient  of  resistance  in  pipe  thus 
coated  is  about  n=.000325  in  terms  of  d  in  feet.  The  average 
value  of  n  for  common  cast  iron  pipe  while  clean  is  n=.0003938 
in  terms  of  d  in  feet. 

Ordinary  lead  pipe  gives  m=.000135  in  terms  of  d  in  feet, 
or  0=86.07.  In  terms  of  r  in  feet,  ordinary  lead  pipe  gives 
m=.000016875,  or  C=243.20.  Lead  pipe  varies  in  specific  grav- 
ity, and  the  coefficient  varies  with  the  specific  gravity.  Very 
dense,  smooth  lead  pipe  gives  values  of  C  in  terms  of  r  as  high 
as  C=297.00  before  the  pipe  becomes  incrusted  or  scaled. 


58  SULLIVAN'S  NEW  HYDRAULICS 

13— Value  of  C  Where  the  Flow  is  in  Contact  with  Dif- 
ferent Classes  of  Perimeter  at  the  Same  Time.— 

The  sides  of  a  channel  may  be  rough  and  covered  with 
vegetation  while  the  bottom  is  smooth  and  clean.  In  such 
case  the  value  of  C  will  decrease  as  depth  of  flow  increases, 
because  of  the  gain  in  ratio  of  rough  to  smooth  perimeter  as 
depth  increases.  On  the  contrary  the  bottom  may  be  rough, 
stony  and  irregular,  while  the  sides  are  smooth,  clean  and 
regular.  In  the  latter  case  the  value  of  C  will  increase  as 
depth  of  flow  increases,  because  of  the  gain  in  ratio  of  smooth 
to  rough  perimeter  as  depth  of  flow  increases.  In  all  such 
cases  it  is  necessary  to  arrive  at  the  mean  or  the  average 
roughness  of  the  combined  classes  of  perimeter.  If  the  flow 
is  two  feet  deep  in  a  canal  six  feet  wide  on  the  bottom  and 
the  sides  are  smooth  and  vertical,  while  the  bottom  is  rough 
and  stony,  let  us  suppose  that  the  sides  correspond  with  C 
=60,  and  the  bottom  with  C=30.  Then  we  have  the  two 
smooth  sides  equal  4  feet  and  the  rough  bottom  equal  6  feet 
and  the  whole  perimeter  equal  10  feet. 

Then,— l_=Smooth  perimeter  where  C=60. 

6 
-TQ — =  rough  uerimeter  where  C=30. 

4X60      240  6X30     180 

And~io~=  lo =  24>      ~io-=To~=18-    And    24 

+18=  42. 

The  value  of  C  for  this  combination  of  perimeters  would 
be  42. 

14.— Tables  of  Coefficients.— In  the  following  tables 
of  coefficients  as  developed  from  the  published  data  of  exper- 
iments, the  groups  are  arranged  with  reference  to  smoothness 
or  roughness  of  wet  perimeter.  The  remarks  in  regard  to  the 
available  data  for  this  purpose,  which  were  made  in  the  in- 
troductory to  this  volume,  should  not  be  forgotten.  Only  a 
part  of  the  available  data  have  been  used,  and  that  was  sim- 
ply a  choice  between  evils  in  many  cases.  The  writer  is  in- 


SULLIVAN'S  NEW  HYDRAULICS. 


debted  to  Mr.  Charles  D.  Smith,  C.  E.,  of  Visalia,  California, 
for  the  data  of  the  guagings  by  him  of  sixteen  canals  in  the 
vicinity  of  Visalia,  California.  It  is  believed  that  these  data, 
all  of  which  are  given  the  common  name  of  "Visalia  Canal," 
are  good  and  reliable.  The  writer  is  also  indebted  to  Mr.  J. 
T.  Fanning  for  a  diagram  of  the  results  of  experiments  by 
him  on  cast  iron  pipes  of  diameters  ranging  from  4  inches 
to  96  inches,  and  exhibiting  the  average  value  of  the  coeffi- 
cient in  such  pipes;  and  for  guagings  of  the  New  Croton 
aqueduct  recently,  made  by  Mr.  Pteley,  and  for  numerous 
valuable  suggestions.  The  writer  is  indebted  to  Mr.  Otto 
Von  Geldern,C.  E.,  of  San  Francisco,  for  the  guagings  of  the 
Sacramento  river  by  C.  E.  Grunsky,  C.  E. 

GROUP  No.  1,  STRAIGHT  LEAD    PIPE.      (Rennie.) 


L'GTH 

DIAM. 

S 

V 

COEFFICIENT 

COEFFICIENT 

yd' 

FEET 

FEET 

SLOPE 

FEET 

m  ^Xi/d3 

'c  r* 

FEET 

~VSl/d3 

•"15:00 

0.0417 

.26666 

5.00 

.0000908 

105.00 

.008515 

Straight  lead  pipe.    (W.  A.  Provis.) 


100.00 

0.125 

.02917 

3.09 

.0001350 

86.07 

.04119 

80  00 

0  125 

.03646 

H.396 

.'  001397 

81.60 

0*419 

60.00 

0.125 

.04861 

3.903 

.0001410 

84.21 

.04419 

The  coefficients  for  pipes  are  in  terms  of  diameter  in  feet. 
Straight  Lead  Pipe.          (W.  A.  Provis.) 


L'GTH 

DIAM. 

S 

V 

COEFFICIENT 

COEFFICIENT 

/d8 

FEET 

FEET 

SLOPE 

FEET 

S 

i  

FEET 

SEC. 

m=  V,XV  d3 

'=^S/d3 

40. 
20. 

0.125 
0.125 

.07292 
.14583 

4.759 
6.150 

.0001422 
.0001703 

83.  b6 
76.55 

.04419 
.04419 

REMARK. — The  coefficient  m  or  C ,  includes  all  resistances 
to  flow,  including  the  resistance  to  entry  into  the  pipe.  In 
such  very  short  pipes,  where  the  velocity  is  considerable,  the 
effect  of  resistance  to  entry  will  materially  affect  the  coeffi- 
cient. For  this  reason  a  general  pipe  formula  for  ordinary 
lengths  of  pipe  will  not  apply  with  accuracy  to  short  tubes 
or  very  short  pipes.  A  special  formula  for  short  pipes  or 
tubes  should  be  applied  in  such  cases.  It  is  not  known 


60 


SULLIVAN'S  NEW  HYDRAULICS. 


whether  all  the  above  lead  pipes  of  different  lengths  were  of 
the  same  quality  and  in  the  same  condition  or  not.  It  is 
probable  that  they  were,  and  that  the  decrease  in  length 
of  pipe  and  increase  in  velocity  greatly  affected  the  resist- 
ance to  entry.  The  resistance  to  entry  of  a  pipe  cut  off 
square  and  flush  with  the  inner  walls  of  the  reservoir  is  al 
ways  equal  to  .505  of  the  head  generating  the  velocity  of 
flow  through  such  pipe.  Hence  in  order  to  obtain  the  true 
coefficient  of  flow  due  only  to  the  resistance  of  the  inner  cir 
cumference  of  the  pipe,  the  entry  head  should  first  be  de- 

v8 
ducted.     The   entry  head=-^-X-505. 

The  data  of  experiments  on  very  short  pipes  are  not  re- 
liable, and  should  never  be  relied  upon.  They  have  no  appli- 
cation to  long  pipes. 

Lead  Pipe  —  (Iben)  Example  of  erroneous  data. 


L'GTH 

FEET 

TOTAL 
HEAD 
FEET 

DIAM. 
FEET 

!/d« 

FEET 

AL- 
LEGED 
VELOC 
ITIES 

COEFFICIENT 

s 
m=VirXi/d3 

COEFFICIENT 

°=A/—  ~ 

^S/  d8 

350.30 
350.30 

17.71 
122.01 

0.082 
0.082 

.02384 
.02384 

2.70 
9.11 

.000162^4 
.0001000447 

78.36 
99.97 

REMARK— Here  are  the  alleged  results  of  two  experiments 
on  the  same  pipe — the  only  difference  in  conditions  being  a 
change  of  head.  As  the  length,  diameter  and  roughness 
were  absolutely  the  same  in  both  cases,  the  only  possible  ef- 
fect of  varying  the  head  would  be  that  the  velocity  would 
vary  directly  as  the  square  root  of  the  head  varied,  and 
nothing  else. 

Where  all  the  other  conditions  are  constant,the  velocity  will 
vary  directly  as  the  square  root  of  the  head>  and  the  resist- 
ance, or  loss  of  head  by  friction,  will  vary  directly  as  the 
square  of  the  velocity.  If  this  is  not  true,  then  the  law  of 
gravity  and  the  law  of  friction  as  accepted  by  the  scientists 
are  necessarily  erroneous,  and  all  scientific  calculations  based 
upon  those  laws  must  fail. 

In  the  first  experiment  with  this  pipe  of  constant  length, 
diameter  and  roughness,  the  head  was  17.71  feet,  and  velocity 
was  2.70  feet  per  second.  As  all  conditions  remained  constant 
except  an  increase  in  head,  then  by  the  law  of  gravity  and  of 


SULLIVAN'S  NEW  HYDRAULICS.  61 

friction  we  would  have 

V/H  :   ^/H   ::  v    :  v;  or  4.148  :  11.08  ::  2.70  :  721 
In  the  last  experiment,  Iben    makes  v==9.11  instead  of 
7.21. 

If  the  velocity  was  correct  in  the  first  experiment,  or 
v=2.70,  then  the  head  lost  by  friction  for  this  velocity  was 
equal  to  the  total  head  minus  the  head  which  remained  to 
generate  the  2  70  feet  velocity.  The  head  required  to  generate 

2.70  feet  per  second  velocity  was  hv=    V*  =(2  70)2  =0.1132  ft. 

64.4      64.4 

The  head  lost  by  friction  at  this  velocity  was   therefore   17.71 
—.1132=17.59  feet,  and  vs=7.29.    Now,  if  the  law  of  friction 
is  correct,  to  wit,  that  friction  will  increase  in  a  constant  di- 
ameter and  length  as  the  square  of  the  velocity,  then  the  loss 
of  head  in  feet  by  friction  in  this  pipe  when    the   velocity   in- 
creased to  9  11  feet  per  second,  would  be 
v8  :  v8  ::head  lost  :  head  lost,  or 
7.29  :  83.00::  17.59  :  211.00. 

In  other  words  in  Iben's  second  experiment  where  the 
total  head  was  only  122.00  feet,  he  was  able  to  lose  211.00  feet 
by  friction,  and  still  have  remaining  1.29  feet  head  to  generate 
the  9.11  feet  per  second  velocity,  which  is  alleged  to  have  oc- 
curred. It  is  conclusive  that  the  laws  of  friction  and  of  grav- 
ity are  absurd,  or  such  data  are  in  error. 

All  correct  experimental  data  of  flow  for  the  same  length, 
diameter  and  roughness  of  pipe  will  necessarily  develop  the 

same  value  of  either  of  the  coefficients,  n,  m  or  C,   regardless 

o 

of  all  changes  in  head  or  velocity,  because  the  ratio— -^  is  nec- 
essarily constant  in  any  given  pipe.  The  foregoing  illustra- 
tration  is  given  as  a  suggestion  of  a  correct  method  of  testing 
the  value  of  such  published  data  of  flow  as  are  now  available. 
Most  of  such  data  are  furnished  by  experiments  of  a  century 
or  more  ago,  and  have  been  translated  from  one  language  to 
another  and  reduced  from  one  system  to  another,  and  printed 
and  reprinted  until  the  accumulated  errors,  added  to  the 
original  crude  methods  in  vogue  a  century  ago,  render  them 


62  SULLIVAN'S  NEW  HYDRAULICS. 

of  very  uncertain  value.  The  writer  is  aware  that  Panning 
and  other  very  eminent  hydraulicians  have  been  of  opinion 
that  m  will  decrease  or  C  increase  with  the  velocity  in  a 
constant  diameter,  but  this  theory  is  not  sustained  by  the  re- 
sults of  Fanning's  experiments  on  a  constant  diameter  (See 
Group  No.  4)  nor  by  the  results  of  experiments  by  the  writer 
(Group  No.  3).  That  theory  cannot  be  accepted  without  first 
rejecting  the  law  of  gravity  and  of  resistance  as  now  generally 
accepted.  If  C  increases  with  an  increased  velocity  in  a  con- 
stant diameter,  it  is  obvious  that  resistance  does  not  increase 

S 
as  rapidly  as  v8,  and  hence  the  ratio—y  would  not  be  constant 

but  would  vary  with  the  velocity.  If  that  is  true,  then  v*  = 
2gH — the  fundamental  law  of  gravity — is  necessarily  untrue, 
and  all  our  learned  discussions  of  equilibrium  and  of  uniform 
flow  are  mere  theoretical  myths  and  rubbish.  Either  that 
theory  or  the  law  of  gravity  and  resistance  must  be  rejected, 
for  both  cannot  stand.  The  experimental  data  now  available 
afford  as  much  evidence  to  sustain  an  opposite  theory  as  to 
sustain  the  above  theory,  and  hence  these  opposite  results 
destroy  both  theories,  and  prove  only  the  erroneousness  of  the 
data.  The  evidence  to  sustain  one  theory  destroys  that  which 
sustains  the  opposite  theory,  and  the  laws  of  gravity  and  of 
resistance  positively  refute  both  theories,  and  establish  the 
theory  that  m  or  C  is  constant  for  all  velocities  in  a  constant 
diameter,  except  as  slightly  affected  by  the  resistance  to 
entry  into  the  pipe.  If  the  entry  to  the  pipe  is  in  the  form  of 
the  vena  contraota,  then  the  velocity  cannot  affect  the  value 
of  C  or  m  at  all. 


SULLIVAN'S  NEW  HYDRAULICS. 


63 


GROUP  No.  2. 
Asphaltum  Coated  Pipe. 


COATED  PIPES. 
[Hamilton    Smith  Jr.] 


Lgth. 
Feet 

Diam. 
Feet 

v/d3 
Feet 

S 

Slope 

V 

Feet 
Sec. 

Coefficient. 
S/d* 

m—       y8 

Coefficient 

C        P~ 
A/S^/d8 

1200.00 

2.154 

3J61 

.01641 

12.605 

.0003265 

55.34 

700.00 

1.056 

1.085 

.00668 

4.595 

.0003432 

54.00 

700.00 

1.056 

1.085 

.01428 

6.962 

.0003200 

55.90 

700.00 

1.0b6 

1.085 

.02219 

8.646 

.0003220 

55.73 

700.00 

1.056 

1.085 

.03319 

10.706 

.0003142 

56.40 

4440.00 

1.416 

1.685 

.06672 

20.143 

.0002771 

60.07 

700.00 

0.911 

0.8695 

.0085 

4.712 

.0003330 

54.80 

700.00 

0.911 

0.8695 

.01334 

6.094 

.0003123 

56.58 

700.00 

0.911 

0  8695 

.01695 

6.927 

.0003072 

57.05 

700.00 

0.911 

0.8695 

.02559 

8.659 

.0003000 

57.73 

700.00 

.230 

1.364 

.01097 

6.841 

.00032000 

55.90 

700.00 

.230 

1.364 

.01227 

7.314 

.00031264 

56.56 

700.00 

.230 

1.364 

.01646 

8.462 

.000313.^6 

56.48 

700.00 

.230 

1.364 

.02470 

10.593 

.00030025 

57.71 

700.00 

.230 

1.364 

.03231 

12.090 

.00030150 

57.58 

REMARK. — The  slight  variation  of  C  or  m  in  the  same 
diameter  and  length  is  due  to  errors  in  weir  or  orifice  coeffi- 
cients used  in  determining  the  velocities.  The  above  pipes 
were  double  riveted  lap  seam  wrought  iron  pipes  put  together 
like  stove-pipe  joints.  Some  of  the  velocities  were  determin- 
ed by  weir  and  others  by  orifice  measurement.  The  differ- 
ence in  value  of  C  for  different  diameters  is  due  to  difference 
in  quality  of  the  coating.  (See  §  12).  In  applying  the  above 
coefficients  it  should  be  remembered  that  these  pipes  were 
new  and  laid  straight,  and  had  free  discharge  and  high  veloci- 
ties which  would  prevent  any  deposit  in  them.  The  propor- 
tion of  asphaltum  in  the  coating  is  not  stated.  This  is  im- 
portant and  should  be  known. 

Cast  Iron    Asphaltum    Coated  Pipe.— [Lampe]. 


Legth 
Feet. 

Diam. 
Feet 

V/d3 
Feet 

S 

Slope. 

V 

Feet 
Sec. 

Coefficient 
S/d3 

Coefficient 

r~v*~ 
P.  ./  v 

™-    v* 

"Vs^/d* 

26,000 
26,000 
26000 
26.000 

1.373 
1.373 
1.373 
1.373 

1.609 
1.609 
1.609 
1.609 

.000594 
.001376 
.00163 
.00195 

1.577 
2.479 
2.709 
3.090 

.0003840 

.(1003601) 

.0003574 

.0003300 

51.03 
52.69 
52.91 
55.04 

REMARK. — This  pipe  had  been  in  use  five  years.  Velocity 
was  judged  of  by  reservoir  contents  and  pressure  guage. 
The  last  coefficient  is  probably  the  true  one.  As  the  veloci- 
ties tabled  in  the  constant  length  and  diameter  do  not  cor- 
respond with  the  slopes  tabled,  it  is  impossible  to  ascertain 
whether  either  of  the  coefficients  are  correct  or  not.  Only 


SULLIVAN'S  NEW  HYDRAULICS. 


one  of  them  can  be  correct.     The  last  one  IB  about  the    aver- 
age value  of  the  coefficient  for  such  coated  pipes. 
Cast   Iron  Asphal turn  Coated  Pipe.     [D'Arcyl. 


L'gth. 
Feet 

Diam. 
Feet 

1/d* 
Feet 

S 

Slope 

V 

Feet 
Sec 

Coefficient 
Sv/d" 

Coefficient 

C=   IVv* 

V* 

^Sy^d8 

365.00 
365.00 
365.00 
365.00 
365.00 

0.6168 
0.6168 
0.6168 
0.6168 
0.6168 

0.4844 
0.4844 
0.4844 
0.4844 
0.4844 

.00027 
.00368 
.02250 
.10980 
.14591 

0.673 

2.487 
6.342 
14.183 
16.168 

.00028SO 
.0002882 
.0002710 
.0002644 
.0002704 

58.82 
58.90 
60.74 
61.50 
60.95 

REMARK. — Velocities  determined  by  orifice.  Variation 
in  C  is  due  to  error  in  orifice  coefficients  used.  This  pipe  was 
quite  short,  and  must  have  had  a  remarkably  smooth  coating. 
The  coefficients  developed  by  this  pipe  are  too  high  for  safe 
use  in  ordinary  practice.  Lap  welded  wrought  iron  pipe  in 
long  lengths  with  few  joints,  when  coated  with  asphaltum 
and  oil,  give  C=60.00.  It  will  be  noted  that  D'Arcy's  data 
generally  give  the  value  of  C  too  high.  As  would  be  expect- 
ed from  a  series  of  experiments  especially  planned  with  ref- 
erence to  the  most  favorable  conditions. 

The  weir  and  orifice  coefficients  should  be  standardized 
in  the  same  manner  as  m  or  C,  so  that  a  given  form  of  weir 
or  orifice  would  have  a  unit  coefficient  which  would  vary 
with  i/rs  for  any  dimensions  of  weir  notch  or  orifice.  The 
results  would  then  be  uniform  and  correct.  

Such  weir  formula  might  take  the  form,  q=A  ~5~-\/ 

"  *       m 

The  value  of  m  would  depend  upon  the  form  of  the  weir 
only,  and  would,  apply  to  all  dimensions  of  weirs  of  that 
given  form.  Before  this  kind  of  a  weir  formula  could  be 
successfully  adopted,  however,  it  would  be  necessary  to  so 
construct  the  weir  as  to  suppress  all  contraction  of  the  dis- 
charge, for  the  contraction  seems  to  follow  no  law.  (See 
Apendix.) 

The  Loch  Katrine  Cast  Iron  Pipe .     Coated  with  Dr.    Smith's 
Coal   Pitch.  [Gale]. 


Lgth 

3M 
Miles 

Diam. 

Feet 

/d« 
Feet 

S 
Slope 

V 
Feet 
Sec. 

Coefficient 
ST/d8 

H):=:  »"  

Coefficient 

c  \!   v" 

^Sv/ds 

33£  m. 

4.00 

8.00 

.000947 

3.458 

.0006344 

39.70 

This  pipe  probably  had  large  deposits  of  gravel  in  it.     It 


SULLIVAN'S  NEW  HYDRAULICS. 


was  evidently  very  rough  from  some  cause.  We  give  its 
coefficient  here  simply  because  this  particular  pipe  has  been 
the  subject  of  so  much  discussion.  See  Flynn'e  "Flow  of 
Water,"  page  34,  for  remark  of  Rankine  and  Humber  on  this 
pipe. 

GROUP  No.  3. 
Clean  cast  iron  pipes— not  coated.    (See  §  12.)    (Sullivan.) 


L'gth 
Feet 


2,800 
2,800 


Diam. 
Feet. 


Feet 


8 

Slope 


Feet 
Sec. 


1.648 
2.771 


3.296 
5.540 


Coefficient 


Coefficient 


52.08 
52.11 
52.10 
52.11 
52.11 


REMARK. — These  experiments  were  the  foundation  of  the 
writer's  formula.  They  were  made  with  the  greatest  possible 
care.  The  writer  being  aware  that  a  weir  or  orifice  coeffici- 
ent determined  by  the  use  of  one  degree  of  convergence  of  the 
edges  of  the  plate  would  not  apply  to  another  degree  of  con- 
vergence or  divergence,  and  having  discovered  discrepancies 
of  several  per  cent,  in  velocities  thus  determined, 
did  not  rely  on  such  measurements  in  the  above  ex- 
periments, but  erected  a  large  measuring  tank  into  which 
the  pipe  discharged.  The  velocities  were  then  determined 

by  the  formula  v=cubic  feet  8ecopd.    The  pipes  were  remark- 
area  in  sq.  feet 

ably  dense  and  smooth,  and  had  never  before  been  wet. 
They  were  laid  straight  and  perfectly  jointed.  In  doubling 
the  diameters  and  increasing  the  head  four  times,  as  will  be 
observed  in  the  above  table,  it  was  the  purpose  to  test  the 
law  of  gravity  as  well  as  to  test  the  effect  upon  the  flow  of 
doubling  the  diameter  while  the  head  remained  constant. 
A  study  of  the  results  thus  obtained  resulted  in  the  form- 
ula for  flow  herein  presented. 

It  may  be  remarked  here  that  the  coefficients  developed 
by  the  experiments  under  these  exceedingly  favorable  cir- 
cumstances with  absolutely  clean,  very  dense,  straight  pipes, 
are  not  to  be  relied  on  for  average  weight  cast  iron  pipes  laid 
in  the  ordinary  manner.  For  average  weight  new  cast  iron 
pipe,  as  long  as  it  remains  clean,  m=.0004,  and  C=50. 

The  nature  of  the  water  which  flows  in  a  pipe  which  is 
not  coated  may  materially  roughen  the  walls  and  reduce  the 


GO 


SULLIVAN'S  NEW  HYDRAULICS. 


value  of  the  coefficient  in  a  very  short  time.  Allowance  should 
always  be  made  for  this  deterioration  by  adopting  diameters 
amply  large. 

GROUP  No.  4. 
Cement  mortar  lined  wrought  iron  pipes, — (Fanning.) 


L'gth 

Diam- 

yd8 

S 

V 

Coefficient 

Coefficient 

Feet 

eter 
Feet 

Feet 

Slope 

Sec. 

o~i/  d" 

C        / 

va 

8171  00 

1.667 

2.153 

.00044 

1.488 

.0004300 

48.22 

8171.00 

1.667 

2.153 

.00073 

1.925 

.0004241 

48.56 

8171.00 

1.667 

2.153 

.00104 

2.329 

.0004130 

49  20 

8171.00 

1.667 

2.153 

.00134 

2.598 

.0004274 

48.38 

8171.00 

1.667 

2.153 

.00158 

2.867 

.0004139 

49.15 

8171.00 

1.667 

2.153 

.00199 

3.271 

.0004004 

49  97 

8171.00 

1.667 

2.153 

.00228 

3.439 

.0004151 

49.08 

8171.00 

1.667 

2.153 

.00272 

3.741 

.0004183 

48.92 

8171.00 

1.667 

2.153 

.00300 

3.920 

.0004203 

48.78 

8171.00 

1.667 

2.153 

.00313 

4.000 

.0004212 

48.72 

8171.00 

1  667 

2.153 

.00320 

4.040 

.0004221 

48.67 

REMARK. — This  was  a  force  main,  and  velocities  were 
measured  at  the  pump.  Considering  slight  errors  in  calcula- 
tions of  slip,  it  is  seen  how  nearly  constant  the  coefficients 
are.  If  there  were  no  errors  of  slip,  &c.,  there  would  result 
but  one  constant  value  of  m  and  C  throughout,  The  above 
guagings  were  remarkably  accurate  if  the  conditions  under 
which  they  were  made  be  considered.  They  show  great  care 
and  excellent  judgment  on  the  part  of  the  experimentalist. 
Under  more  favorable  conditions,  still  closer  results  would 
have  been  had.  From  the  values  of  the  coefficient  it  is  prob- 
able that  the  lining  of  this  pipe  was  one  third  sand  and  two- 
thirds  cement.  Neat  cement  linings  develop  higher  values  of 
C  than  the  above,  while  the  above  coefficients  agree  closely 
with  those  for  linings  of  one-third  sand  and  two-thirds  ce- 
ment. 

The  value  of  C  does  not  increase  with  an  increased  veloc- 
ity in  a  constant  diameter,  as  has  been  claimed  by  some  au- 
thors. If  so,  the  last  value  of  C  in  the  above  table  should  be 
the  greatest. 


SULLIVAN'S  NEW  HYDRAULICS. 
GROUP  No.  5. 


67 


Wooden  conduits,  planed    poplar,     closely  jointed.     (D'Arcy 
&  Bazin.) 


L'gth 
Feet 


Feet 


J/R3 

Feet 


S 
Slope  Feet  Sec, 


230.58 
230.58 
280.58 
230.58 


230.58 


0.505 
0.5C5 
0.505 
0.505 

o.r,or, 
0.  :,(>:, 
o.-,o:, 
0.505 


.000475 
.001076 


<  ).:<>!  i 


.002911 
.00(072 


.00576 
.006614 


1.666 
2.519 
3.372 
4.225 
5.068 
5.527 
5.914 
6.373 


.00006143 


.00005S53 

.(KKHWITO 
.00005948 

.(MHKMilK.KI 

.oooo:>.vi:> 


129.10 
130.60 


129.75 
129.10 
130.70 


REMARK.— This  conduit  had  a  bottom  width  of  2.624  feet 
and  was  1.64  feet  in  depth.  The  velocities  were  determined 
by  weir  measurement.  The  values  of  C  developed  illus- 
trate the  uncertain  application  of  weir  coefficients  even  in 
the  same  small  channel  and  for  small  differences  in  head,  and 
when  applied  by  persons  of  great  experience  and  sound  judg- 
ment. The  value  of  the  true  coefficient  in  this  conduit  was 
probably  C=129.00  in  each  case.  The  value  of  the  coefficient 
for  planed  wood  surfaces  will  doubtless  vary  with  the  density 
of  the  wood.  The  coefficient  will  be  greater  in  conduits  in 
which  the  boards  are  laid  parallel  to  the  flow  than  where  the 
flow  is  across  the  grain  of  the  wood  and  the  joints.  Assum- 
ing that  m=.00006  is  the  true  coefficient  in  terms  of  r  in  feert 
for  planed  hard  wood  surfaces,  we  may  reduce  to  terms  of  d 
in  feet  (See  §  10)  by  multiplying  by  8,  and  we  have  m=.00048 
or.  C=45.64  in  terms  of  d  in  feet.  This  permits  of  a  direct 
comparison  of  the  relative  degrees  of  resistance  to  flow  in 
wooden  pipes  of  planed  staves  closely  jointed,  and  in  iron 
pipes,  Thus 

Lead  pipes— C=85.00  } 

Asphaltum  coated   pipes,  C=56.00 

Clean  cast  iron  pipes,  C=50.00          ^All  in  terms  of  diam- 

Clean  planed   hard  wood,  C=45.64  ]      eter  in  feet. 

Cement  (one  third  sand)— C=48.50  I 


SULLIVAN'S  NEW  HYDRAULICS. 

Wooden  conduits,     Planed  boards.     (D'Arcy  &  Bazin) 


Surface 
Width 
Feet 

R 

Feet 

v/R3 
Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 
m_Sv/71T 

Coefficient 

n_   /    v" 

v* 

^S,/^ 

3.16 
3.62 
3.89 
4.08 
4.53 
4.59 

0.390 
.537 
.632 
.717 
1.015 
1.148 

0.24355 
.39350 
.50240 
.60710 
1.0225 
1.2300 

.0015 
.0015 
.0015 
.0015 
.0015 
.0015 

2.61 
3.23 
3.71 
4.04 
5.00 
5.54 

.00005363 
.00005660 
.00005475 
.C0005580 
.00006131 
.00006011 

136.55 
132.95 
135.35 
134.80 
127.70 
128.90 

REMARK. — The  velocities  in  this  table  were  determined  by 
surface  floats  and  Pitot-D'Arcy  tube  measurements.  The  ve- 
locities thus  determined  are  undoubtedly  too  high.  The  weir 
measurements  given  in  the  preceding  table  are  more  nearly 
correct.  A  large  majority  of  the  guagings  by  D'Arcy  and 
Bazin  were  made  by  surface  float  and  Pitot  tube  measure- 
ments of  velocity.  They  are  not  reliable  when  so  made.  This 
table  is  introduced  here  to  show  that  velocities  thus  deter- 
mined are  too  high,  and  the  fluctuating  values  of  C  show  that 
this  method  of  guaging  is  not  at  all  reliable.  Data  of  flow  de- 
termined by  such  methods  should  be  avoided.  It  is  not  in- 
tended to  convey  the  idea  that  all  of  D'Arcy  and  Bazin's 
guagings  are  unreliable,  but  to  show  that  such  guagings  as 
are  made  by  surface  floats  or  by  Pitot  tube  are  worthless, 
whether  made  by  them  or  any  one  else.  Some  of  D'Arcy's  data 
are  good.  Actual  tank  measurement  of  the  discharge  is  the 
only  really  accurate  method  of  determining  the  velocity  which 
has  so  far  been  adopted.  Weir  measurement  can  be  made 
accurate  by  adopting  unit  coefficients  for  weirs  similar  to  mor 
C  as  suggested  in  a  remark  under  Group  No,  2,  and  the  Ap- 
pendix I. 

Uiiplaned  boards,  well  jointed  and  without  battens. 

The  average  value  of  m  =  . 000070  in  terms  of  r  in  feet. 
C=119.60    in  terms  of  r  in  feet. 


Ordinary  Flume  6X«5  feet— Straight. 


Clarke 


Length. 
Feet. 

R 

Feet 

jiA8 

jFeet 

S 
Slope 
.000435 

\Feet 
jSec 

Coefficient 

i  Coefficient 

!  c    p1" 

Vs 

'     VSl/r3 

2500. 

1.45 

jl.746 

.000088        ;           106.30 

sewage.    The  grease  and  slime  may  affect  the  flow    consider- 
ably, ae  well  as  the  solid  matter  mixed  with  the  sewage. 


SULLIVAN'S  NEW  HYDRAULICS.  69 

Bough  Irrigation  Flume.  Highline  Flume,  Colorado.  (Wilson) 


Length 
Feet 

R  [v'r8 
Feet  jFeet 

S           V 

Feet 
Slope  iSec. 

Coefficient 
v*~~ 

ICoefficient 

3000 

4.50  J9.546 

00099432:6.7657 

.00020733 

j             69.50 

REMARK— This  is  a  rough  bench  flume  with  many  abrupt 
bends.  For  a  cut  and  description  of  this  flume  see  "Irriga- 
tion Engineering"  by  Herbert  M.  Wilson,  C.  E.,  pages  173 
and  174.  The  bends  reduce  the  value  of  C  considerably  be- 
low its  value  for  a  straight  flume. 

GROUP  No.  6, 

Stone  and    brick  lined   Channels.— Chazilly  Canal.    D'Arcy 
and  Bazin. 


Depth 

R 

g 

Coefficient 

Coefficient 

V'iath 

Feet 

m-8^'3 

r     /   v' 

Feet 

Feet 

Feet 

Slope 

Sec. 

Vs 

-Vsv/7T- 

4.04 
4.10 
4.14 

4.18 

0.50 
0.70 
1.00 

1.20 

0.41 
0.57 
0.68 
0.77 

.0081 
.0081 
.0081 
.0081 

5.73 

7.52 
8.19 
8.75 

.000064765 
.OOOOtJ]i!34 
.OU0067713 
.0  0071483 

124.29 
127.37 
121.52 
118.30 

REMARK  —  This  canal  is  lined  with  smooth  ashlar  or  cut 
stone.  The  gaugings  wera  probably  by  surf  ace  floats  or  Pitot 
tube  which  accounts  for  the  fluctuating  values  of  C  developed. 
If  this  is  not  the  true  cause,  then  the  bottom  and  the  sides 
to  a  depth  of  .70  feet  must  be  very  much  smoother  than  the 
walls  are  above  that  depth.  The  last  value  of  C  is  probably 
nearest  the  correct  value.  See  §  13. 


Roquefavour    Aqueduct. 
sides.    {D'Arcy  &  Bazin.) 


Neat    cement    bottom.     Brick 


Surface 
Width 
Feet 

Depth 
Feet 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 

Sy'r* 

Coefficient 
C-   /    v* 

m  Y—  
va 

A/Sv/  r* 

7.40 

2.50    1  1.504 

.00372 

10.26 

.(WOJ652 

123.85 

REMARK. — This  aqueduct  is  nearly  rectangular  and  at  this 
depth  of  flow  the  smooth  cement  bottom  forms  more  than 
half  the  wet  perimeter.  It  should  therefore  develop  a  greater 
value  of  C  than  the  stone  lined  Chazilly  canal  of  the  preced- 
ing table.  In  a  smooth  bottomed  canal  similar  to  this  aque- 
duct where  the  bottom  is  much  smoother  than  the  sides,  the 


70 


SULLIVAN'S  NEW  HYDRAULICS. 


value  of  C  should  be  greatest  for  the  least  depths  of  flow,  be- 
cause  as  depth  increases  the  proportion  of  the  rougher  side 
perimeter  becomes  greater. 

Aqueduct  de  Crau.    Hammer  dressed  stone.    (D'Arcy  & 
Bazin. 


Surface 
Width 
Feet 

Depth 
Feet 

R 
?eet 

S 

Slope 

V 

Feet. 
Sec. 

Coefficient 
tn     S'/TF 

Coefficient 

C-J    V* 

V* 

^Sv/  r" 

8.50 

3.00 

1.774 

.00084 

5.55 

.0000668 

122.57 

Sudbury  Conduit.    Hard  brick,   well  jointed.    (Fteley  & 
Stearns,  1880.) 


L'gth 
Feet 

Greatest 
Depth 
Feet 

R 

Feet 

S 
Slope 

Feet. 
Sec. 

Coefficient 
m-S^ 

Coefficient 

C-   /    v* 

V* 

W3 

4,200 

1.518 
2.037 
2.519 
3.561 

1.078 
1.385 
1.628 
2.049 

.0001928 

.noiniej 

.  00019  M 
.1001929 

1.827 
2.139 
2.372 
2.72J 

.00006460 
.00006851 
.00007115 
.00;  07648 

124.33 
120.82 
118.60 
114.35 

REMARK. — Velocities  measured  by  weir.  Only  four  of 
these  guagings  are  given  because  the  slopes  of  water  surface 
in  the  others  are  so  different  from  the  slope  of  the  conduit 
and  from  each  other  as  to  show  that  equilibrium  and  uniform 
flow  had  not  ensued  when  the  guagings  were  made.  The  co- 
efficients are  remarkably  high  for  a  plain  brick  perimeter. 
The  silt  deposit  on  the  bottom  also  affects  the  flow. 


New     Croton 
(Fteley,  1895.) 


Aqueduct.      Hard     brick,    well    jointed 


Depth 

Area 

R 

S 

V 

Coefficient 

Coefficient 

above 

So 

Feet 

O      /       3 

/        V2 

center  of 
Invert. 

oq. 

Feet 

Feet 

Slope 

Sec. 

V-2 

\Sv/r3 

1.10 

9.24 

0.7434 

00013257 

1.0969 

.0000706267 

118.97 

1.50 

14.12 

1.0656 

.00013257 

1.4338 

.0000709351 

118.71 

.     2.10 

21.57 

1.4886 

.00013257 

1.7731 

.0000766000 

114.26 

3.00 

33.04 

2.0236 

.00013257 

2.1281 

.0000842174 

108.97 

4.00 

46.14 

2.5137 

00013257 

2.4102 

.0000906000 

105.08 

5.20 

62.20 

2.9947 

00013257 

2.6560 

.0000974027 

101.35 

6.80 

83.89 

3.4998 

.00013257 

2.8894 

.000103930 

98.09 

9.20 

115.78 

4.0062 

00013257 

3.0989 

.000110689 

95.05 

11.00 

136.93 

4.1417 

.00013257 

3.1519 

.000112500 

94.28 

12.50 

150.55 

4.0031 

00013257 

3.0977 

.000110600 

95.09 

12.842 

152.81 

3.9161 

'00013357 

3.0625 

.000109530 

95.55 

SULLIVAN'S  NEW  HYDRAULICS. 


71 


REMARK.— Mr.  Fteley  states  in  his  report  that  the  veloc- 
ities for  depths  below  1.90  feet  are  not  as  accurate  as  those 
for  greater  depths,  as  the  bottom  or  invert  has  slight  silt  de- 
posits. It  is  evident  that  the  bottom  is  very  much  smoother 
than  the  sides,  or  the  guaging  apparatus  was  greatly  at 
fault.  With  the  assistance  of  a  very  smooth  silted  bottom 
the  side  walls  and  arch  are  apparently  so  rough  as  to  run  the 
value  of  C  below  its  value  for  common  brick  masonry.  This 
is  a  conduit  of  the  horse  shoe  form  and  the  velocities  were 
measured  by  meter.  The  result  does  not  commend  meter 
guagings.  From  the  slope  of  water  surface  it  appears  that 
uniform  flow  was  attained  in  each  case  before  the  guagings 
were  made.  See  §  13. 

First  Class  Brick  Conduits  Washed  Inside  With  Cement.* 
(Fteley.) 


Name  of 
Conduit 

R 

Feet 

S 

Slope 

V 

Feet 
Sec. 

Coefficient 

Coefficient 

•^ 

C=V^ 

Sudbury 
Cochituate 

2.4588 
1.4170 

.00020 
.0000496 

3.029 
1.000 

.000084000 
.000083637 

109.11 
109.34 

*See  "Water  Supply  Engineering"  by  J.  T.  Fanning,   p.   445, 
Ninth  Edition. 


Washington,  D.  C.,  Aqueduct,  Brick  Conduit.    Completed 
1859.    See  Fanning,  P  445.) 


R 

Feet 

S 

Slope 

v 
Feet  Sec. 

Coefficient 
m     Sl/7F 

Coefficient 

c  r^~ 

-^S/T' 

1.8735 

.00015 

1.893 

.000107218 

96.60 

REMARK.— This  is  about  the  correct  value  of  C  for  ordi- 
nary brick  perimeters  after  several  year's  use.  Where  spe- 
cially smooth  or  scraped  brick  are  used  or  a  cement  wash  is 
applied  the  value  of  C  will  be  greater.  Pure  cement  linings 
in  channels  of  uniform  cross  section  and  good  alignment  de- 
velop an  average  value  of  C=150.00  in  terms  of  r  in  feet.  The 
value  of  C  will  vary  somewhat  with  different  qualities  acd 
fineness  of  pure  cement  linings,  and  uniformity  of  the  walls. 


72 


SULLIVAN'S  NEW  HYDRAULICS. 


Sudbury  Conduit. — Hard  Brick  With   Surfaces  Scraped. 
(Fteley  &  Stearns  1880.) 


Greatest 

R 

S 

V 

Coefficient 

Coefficient 

Depth 

Feet 

Slope 

Ft.  Sec. 

Sv/r3 

V* 

0.719 
1.055 
1.076 
1.187 
1.224 
1.328 
1.415 

0.493 
0.762 
0.778 
0.858 
0.885 
0.957 
1.016 

.0001640 
.0001742 
.0000983 
.0000246 
.0001715 
.0000746 
.0000140 

1.079 
1.423 
1.098 
0.550 
1.577 
1.064 
0.443 

.000048763 
.000057200 
.000005950 
.000064450 
.000057600 
.000061700 
.000073000 

143.21 
132.28 
133.65 
124.85 
131.79 
127.75 
117.10 

REMARK.— This  conduit  is  of  the  horse  shoe  form  and  600 
feet  in  length.  Velocities  were  determined  by  weir.  The 
conduit  has  a  grade  S=.00016.  Compare  the  slopes  in  the 
above  table  with  that  of  the  couduit.  Also  compare  the 
depths  of  flow  with  the  corresponding  velocities  tabled.  It 
is  quite  remarkable  to  note  the  great  changes  in  S  for  such 
very  small  changes  in  R  in  a  uniform  channel  with  a  grade 
S=.00016.  As  the  slope  of  water  surface  is  so  different  from 
that  of  the  bottom  of  the  conduit,  it  necessarily  follows  that 
the  depth  of  flow  must  have  been  different  at  each  successive 
point  along  the  conduit,  and  the  value  of  r  was  different  at 
each  different  point.  The  velocities  were  inversely  as  the 
depths  or  wetted  cross  sections  and  hence  were  greatest 
where  the  depths  were  least.  Uniform  flow  had  not  occurred 
and  hence  the  effective  value  of  S  could  not  be  known. 

A  comparison  of  the  values  of  C  for  this  conduit  with 
those  for  carefully  dressed  poplar  conduits  (Group  No.  5)  and 
for  average  weight  clean  cast  iron  pipes  would  show  this 
brick  surface  to  be  smoother  than  either  of  the  others.  This 
is,  of  course,  not  the  fact.  Because  of  the  great  number  of 
joints  and  resulting  small  irregularities  of  a  brick  wall,  it  is 
scarcely  possible  that  such  wall  should  be  more  uniform  and 
smooth  than  a  carefully  constructed  conduit  of  unplained 
boards  of  hard  wood,  unless  the  wall  were  coated.  In  the 
latter  case  the  wetted  perimeter  would  consist  of  the  coating 
and  not  of  brick.  Such  data,  although  from  eminent  author- 
ity, cannot  be  accepted.  The  last  value  is  nearest  correct. 


SULLIVAN'S  NEW  HYDRAULICS.  73 

Brick  lined  channel.  (D'Arcy  and  Bazin) 


Area 

R 

g 

Coefficient 

Coefficient 

m-8^13 

r       /    V" 

Sq.  Feet 

Feet 

Slope 

Feet  Sec. 

v» 

^vi2E 

6.22 

0.7554 

.0049 

6.69 

118.67 

REMARK. — Velocity  determined  by  surface  float  and  Pitot 
tube  which  almost  invariably  gives  the  mean  velocity  much 
too  high.  This  error  results  in  giving  too  great  a  value  to  C 
for  ordinary  plain  brick  perimeters. 


Croton  Aqueduct.    Brick.     Completed  1842. 
Page    445). 


(See  Fanning, 


2.3415       .00021 


.0001677 


77.50 


REMARK.— This  conduit  is  of  the  horse  shoe  form.  It 
probably  contains  deposits  of  gritty  material  which  reduce  C 
to  so  low  a  value. 


Brooklyn  Conduit.     Brick.     Completed  1859. 
Page  445) 


(See  Fanning 


2.5241        .00010 


REMARK. — As  masonry  conduits  are  permanent  invest- 
ments it  is  best  to  adopt  a  coefficient  value  low  enough  to 
allow  for  deposits  and  future  deterioration  of  perimeter. 

Concrete    Conduits. — Old. — Different    stages   of  ruin.     (See 
Fanning,  page  445). 


Name  of 
Conduit 

R 
Feet 

S 
Slope 

V 

Feet  Sec. 

Coefficient 
m=SJAl 

V2 

Coefficient 
r      1    v* 

-Vsv/73- 

94.87 

84.52 
104.77 

117.00 

Metz 
Pont    du 
Gard 
Pont    Pyla 
Mont- 
pellier 

0.915 

1.250 
0.6109 

0.25 

.00100 

.00040 
.00166 

.00030 

2.783 

2.000 
2.950 

0.716 

.00011175 

.00014000 
.00009110 

.00007310 

REMARK. — In  response  to  a  recent  inquiry  of  the  writer 
Mr.  Fanning  states  that  he  visited  these  conduits  a  few  years 
ago  and  that  some  of  them  appeared  to  be  in  excellent  re- 
pair. They  are  constructed  of  hydraulic  concrete,  and  are 
rectangular  in  form. 


74 


SULLIVAN'S  NEW  HYDRAULICS. 


Spillway  of    Grosbois    Reservoir.    Ashlar    laid    in    Cement. 
(D'Arcy   and  Bazin) 


Surface 

Depth 

R, 

s 

v 

Coefficient 

Coefficient 

Width 
Feet 

Feet 

Feet 

Slope 

Feet 
Sec. 

S,/r« 
m=—  — 

C=A!ST^ 

5.98 
6.01 
6.05 
6.07 

0.36 
.55 
.71 

.84 

0.324 
.467 
.580 
.662 

.101 
.101 
.101 
.101 

12.29 
16.18 
18.68 
21.09 

.00012331 
.00012313 
.00012786 
.00012230 

90.05 
90.12 

88.45 
90.42 

Covered  with  a  slimy  deposit. 

Tail   race  Grosbois    reservoir. 
(D'Arcy  &  Bazin.) 


Ashlar   laid   in    cement. 


Surface 
Width 
Feet 

Dep'h 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 
ST/~T» 

Coefficient 

r      i~~** 

m-     v* 

-VSv/rT 

6.00 
6.10 
6.10 
6.10 

0.49 
.77 
.97 
1.16 

0.424 
.620 
.745 
.852 

.037 
.037 
.037 
.037 

9.04 
11.46 
13.55 
15.08 

.00012499 
.00013750 
.00012958 
.00012800 

89.45 
85.28 
87.85 
88.30 

Covered  with  a  light  slimy  deposit.    Joints   partly  damaged. 
Surface  float. 


Grosbois  Conduit.     Horseshoe    form, 
set  in  mortar.     (D'Arcy  &  Bazin.) 


Stone  masonry 


6.46 
6.50 


2.21 
2.75 
3.12 


0.98 
1.29 
1.49 
1.60 


1.32 
1.90 
2.12 
2.47 


.000115118 


86.50 
93.27 


REMARK. — Bottom  is  rougher  than  sides.  No  deposit. 
Joints  not  damaged.  As  D'Arcy  &  Bazin  nearly  always  give 
the  slope  of  the  bottom  of  the  conduit,  it  is  probable  that 
these  guagings  were  made  at  different  places  along  the  con- 
duit, as  the  slopes  are  different.  The  values  of  C  may  be  at- 
tributed to  the  rough  bottom  and  smooth  sides  and  also  to  er- 
rors in  guaging  with  Pitot  tube  and  floats.  For  ascertaining 
the  correct  value  of  C  for  any  given  depth  in  such  channels 
see  §  13. 


Groisbois  Canal. 
(D'Arcy  &Bazin.) 


SULLIVAN'S  NEW  HYDRAULICS,  75 

Roughly   hammered   stone   masonry. 


Sur- 
face 
Wdth 
Feet 

D'pth 

Feet 

R 

Feet 

s 

Slope 

V 

Feet 
Sec. 

Coefficient 

Coefficient 

m           v2 

A/S,/r° 

3.50 
3.50 
3.60 
3.90 

0.90 
1.20 
1.30 
1.60 

0.62 
.71 

.0600 
.0290 
.0141 
.0121 

13.93 
11.23 
8.36 
7.58 

.00015610 
.00013757 
.00015420 
.00017400 

80.04 
85.26 
80.53 
75.82 

REMARK. — From  the  difference  in  slope  it  is  probable 
that  these  guagings  were  at  different  places  where  the  rough- 
ness was  different.  Otherwise  the  guagings  are  at  fault.  C 
should  be  constant,  unless  the  roughness  of  perimeter  was 
different  at  different  depths  of  flow. 

Qrosbois  Canal.  Stone  Masonry.  Broken  Stones  on 
the  Bottom.  (D'Arcy  &  Bazin.) 


Sur- 
face 
Wdth 
Feet 

D'pth 
Feet 

R 

Feet 

S 
Slope 

Feet 
Sec. 

Coefficient 

—  *£- 

Coefficient 

°=Vs^ 

6.80 
6.90 
6.90 
7.00 

1.50 
2.00 
2.40 
2.70 

0.88 
1.23 
1.40 
1.50 

.000648 
.000671 
.000683 
.000683 

1.47 

2.02 
2.34 

2.78 

.00024755 
.00022434 
!<KX)206G 
.0001624 

63.55 
66.76 
69.57 
78.45 

REMARK.— The  effect  of  the  loose,  broken  stones  and  mud 
deposits  on  the  bottom,  is  to  reduce  the  value  of  the  coeffi- 
cient C  in  the  ratio  that  the  mean  of  the  different  degrees  of 
roughness  increases.  If  the  depth  of  flow  were  reduced  to  .50 
foot,  the  value  of  C  would  not  exceed  45,  because  the  rough 
bottom  perimeter  would  controJ.  It  is  probable  that  if  the 
sides  alone  were  considered  apart  from  the  rough  bottom 
the  value  of  C  would  be  90.00.  The  value  of  C  for  different 
depths  of  tlow  in  such  channels  will  be  different  for  each 
depth,  and  may  be  determined  by  the  rule  given  in  §13. 
Compare  with  New  Croton  Aqueduct. 


76 


SULLIVAN'S  NEW  HYDRAULICS. 


Grosbois  Canal. — Stone  Masonry  in  Bad  Order.     (D'Arcy  and 
Bazin) 


Surface 
Width 
Feet 


6.80 
6.90 
7.00 
7.00 


Depth 
Feet 

1.60 
2.40 
2.90 
3.30 


R 
Feet  Slope 


Feet 
Sec. 


Coefficient 


.00017610 
.00014115 


Coefficient 

C=Jo-V 


61.58 
70.95 
75.36 

84.18 


REMARK.— Broken    stones    and     mud     on    the    bottom. 
Sides  fairly  smooth.    Rule  given  in  §  13  applies. 

Solani  Embankment.     Sides    of    Stone    Masonry.    Stepped. 
(Cunningham). 


Surface 
Width 

Depth  |   R 

1  S  !  V 

iFeet 

Coefficient 
Sv/r3 

Coefficient 

Feet 

Feet   iFee 

iSlope  i  Sec, 

ra—     vs 

~\Sv/r3 

150.00 
150.00 
160.00 
164.00 
170.10 

1.50        ;i.69 
2.30        :2.26 
4.10        14.07 
9.10        17.84 
11.00       19.31 

i.  000090  !  0.44 
i.  000148  i  0.87 
i.  000215  i  1.79 
J.UC0215  i  3.43 
-.000227  i  4.02 

.00102128 
.00066400 
.00055090 
.000401162 
.000400944 

31.28 
38.81 
42.61 
49.92 
49.94 

REMARK— This  channel  has  a  bottom  width  of  150  feet 
The  side  slopes  are  of  stone  masonry,  built  in  steps.  The 
steps  are  broken  and  sunken  in  many  places.  The  bottom  is 
of  clay  and  boulders,  very  irregular,  with  bars  of  brick  and 
boulders  built  across  at  frequent  intervals  to  prevent  scour. 
As  depth  of  flow  increases  the  ratio  of  smoother  side  peri- 
meter increases,  and  the  mean  of  the  different  degrees  of 
roughness  becomes  of  a  lesser  degree  of  roughness.  Hence 
the  value  of  C  will  increase  with  each  increase  in  depth. 


GROUP  No.  7,  RUBBLE  AND  RIP  RAP. 

Chazilly  canal.  Bottom  of  earth.  One  side  wall  of 
mortar  rubble  and  the  other  side  wall  of  dry  laid  rubble. 
(D'Arcy  &  Bazin.) 


Surf. 
Wdth 

Feet 

D'pth 
Feet 

R 

Feet 

S 
Slope 

V 

Feet  Sec. 

Coefficient 
m«§4? 

Coefficient 

r      /  v* 

V8 

A/Sv/r* 

8-50 
9.50 
9.80 
10.20 

1.30 
2.00 
2.40 
2.70 

1.00 
1.36 
1.54 
1.67 

.000525 
.000450 
.000462 
.000487 

1.01 
1.38 
1.58 
1.74 

.00051465 
.00037470 
.00035347 
.00034730 

44.08 
51.66 
53.20 
53.66 

SULLIVAN'S  NEW  HYDRAULICS, 


77 


REMARK— If  these  guagings  were  all  made  at  the  same 
point,  the  values  of  C  show  that  the  earth  bottoms  and  rub- 
ble footings  were  much  rougher  than  the  masonry  side  walls 
higher  up.  See  §  13. 

Turlock  canal  rock  cut.  Partly  excavated  in  slate  rock, 
other  parts  of  dry  laid  rubble  and  rip  rap.  ("IrrigationEogi- 
neering"  by  H.  M.  Wilson,  p.  82-83.) 


Surf. 
Wdth 
Feet 

Dpth 
Feet 

R 

Feet 

s 

Slope 

V 

Feet  Sec. 

Coefficient 

SV/TS" 

*=-?r- 

Coefficient 

-^ 

50.00 

10.00 

5.90 

.0015 

7.50 

.000382 

51.18 

This  rock  cut  is  6,200  feet  in  length  and  forms  part  of  the 
Turlock  canal,  California. 


River  Waal.     Gravel  bottom, 
laid  rubble.     (Krayenhoff.) 


Sides   revetted   with   dry 


17.25   11.10   .0001044    3.165   I 

Description  of  this    rubble,etc.    from    Beaumont's    Geology, 
Paris,  1845. 

Head  Race,  Kapnikbanya,  Hungary.    (Rittinger) 
Bottom  is  paved.    Sides  are  of  dry  laid  rubble  not  smooth  as 
bottom. 


Depth 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 

Coefficient 

C       I 

0.26 
0.45 

0.213 
0.344 

.0038 
.0038 

1.37 
1.83 

.000200 
.000229 

70.71 
66.08 

REMARK.— The  value  of  C  for  rubble  will  depend  on  the 
size  and  shape  of  the  stones.  If  the  stones  are  small  and 
laid  closely  the  coefficient  will  be  much  greater  than  if  large, 
irregular  stones  are  laid  with  large  spaces  and  projections. 

Tail  Race.    Staukau.   Hungary,     Bottom    paved.     Sides    of 
dry  laid  rubble.     (Rittinger) 


Depth 
Feet 

R 

Feet 

S 
Slope 

V 
Feet 
Sec. 

1.257 
1.491 
1.643 

Coefficient 
Sv/r3 

Coefficient 

P       /     V" 

m-     v* 

'-A/Sv/.o 

0.42 
0.56 
0.69 

0.289 
iMr.ii 
0.419 

.0025 
.0025 
.0025 

.0002457 
.0002390 
.0002500 

63.80 
64.73 
63.25 

78 


SULLIVAN'S  NEW  HYDRAULICS. 


Mill  Race.    Pricbram,  Hungary.    Very   rough,   irregular 
bottom  of  earth. 

Side  walls  of  dry  laid  rubble.    Bottom  2.07  feet  width.    (Rit- 
tinger). 


Bottom 
Width 
Feet 

Depth 
Feet 

R 

Feet 

S 

Slope 

V 

Feet 
Sec. 

Coefficient 
Sv/r8 

Coefficient 
r-      /    y8 

-VSv/r" 

2.07 
2.07 
2.07 
2.07 
2.07 
2.07 

0.41 
0.44 
0.70 
0.80 
0.86 
0.90 

0.316 
0.336 
0.472 
0.548 
0.560 
0.566 

.0022 
.0022 
.0022 
.0022 
.0022 
.0022 

0  A  9 

0.588 
1.953 
1.135 
1.190 
I  269 

.00258236 
.001239473 
.0007855758 
.000692858 
.000651000 
.000581962 

19.675 
28.400 
35.670 
37.987 
39.200 
41.450 

REMARK. — This  is  a  good  illustration  of  the  effect  of  a 
combination  of  perimeters  of  different  degrees  of  roughness, 
which  is  referred  to  in  §  13,  The  rough  bottom  and  large 
rough  rubble  footings  at  the  bottom  of  the  side  walls  almost 
prevented  any  bottom  velocity  of  flow. 

Grosbois  Canal. — Rough,  trapezoidal  canal.  The  bottom 
of  earth;  one  Bide  slope  rip-rapped,  the  other  of  earth  with 
some  little  vegetation. 

The   bottom    is    covered    with    stones   and    loose    boulders. 
(D'Arcy  and  Bazin) 


Surface 
Width 
Feet 

Depth 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 
S,/r3 

Coefficient 

c     /  v2 

m—        V2 

ViVr" 

9.10 
11.20 
12.40 
13.40 

1.70 
2.30 
2.60 
2.90 

1.05 
1.37 
1.52 
1.64 

.000936 
.000936 
.000957 
.000964 

1.08 
1.37 
1.56 
1.71 

.00087200 
.00080000 
.00073100 
.00069231 

33.86 
35.36 
36.98 
38.00 

GROUP  No.  8.     IRRIGATION  CANALS. 

A  series  of  Calif ornia  Irrigation  Canals  carefully  guaged. 
By  Charles  D.  Smith,  C.  E.,  of  Visalia,  California.  (1895). 
No.  1.     A  new  canal  in  common  loam  just   completed.     Weir 
measurement. 


R 

Feet 

S 
Slope 

Feet 
Sec. 

Coefficient 
S,/r8 

Coefficient 

c-  /  v8 

m-     v* 

C'  -VSv/r8 

0.99 

.0006 

1.33 

.000334 

54.72 

See  Group  No.  11. 


SULLIVAN'S  NEW  HYDRAULICS. 


79 


No.  2.     An  old  canal  in  common  loam.     In  only  fair  conditi  on. 
By  meter. 


R 
Feet 

S 
Slope 

v 
Feet 
Sec. 

Coefficient 
Sv/r" 

*=-^r- 

Coefficient 

/    v» 

c=VsTvr 

0.85 

.001 

1.52 

.0003392 

54.30 

No.  3,     An  old  canal  in  common  loam   recently   cleaned   but 
not  punned.     By    weir. 


No.  4.    A  canal  in  sandy  gravel,  good   repair.     Firm  gravel. 
By  meter.  


1.40 


4.02 


56.87 


No  5.    A  new  canal.     In  river  sand. 


By  meter. 


1.16 


.00175 


49.83 


Nos.  6  and  7.    Canals  in  clay  with  loose  gravel  on  the  bottom, 
otherwise  clean.     By  meter. 


.00177 
.00194 


3.46 
3.51 


66.04 
64.73 


Nos.  8,  9  and  10.— Canal   in   firm    earth   with   clay    bottoms. 
Good  condition.  By  meter. 


2.00 

.0004 

2.255 

.00022249 

67.08 

3.35 

.00001 

0.531 

.00021740 

67.22 

3.34 

.0000375 

1.032 

.00021486 

68.22 

Nos.  11,  12  and  13. — Canals  in  very  heavy,  smooth  earth, 
recently  cleaned,  trimmed  and  punned  and  put  in  excellent 
order.  Guaged  by  weir. 


.000184553 
.000195000 


Nos.  14  and  15.    Old  canals  grown    up   with  weeds   reaching 
nearly  to  the  surface.  Weir  and  meter, 


1.13 
1.77 


.00060 


0.868 
0.845 


.001154 


No.  16  —Very  crooked  old  slough  in  firm  earth.          By  meter. 

0.91     I    .0030     |   3.086   |   .00027354   |      60.47 


SULLIVAN'S  NEW  HYDRAULICS. 


GROUP.  No.    9.— SMOOTH  CANALS  IN  EARTH. 

Mill  race,  Kagiswyl,  Switzerland.  Side  slopes  of  firm 
earth,  smooth.  Bottom  covered  with  fine  gravel.  Guaged 
by  meter.  (Epper.) 


R 

Feet 

S 

Slope 

Feet  Sec. 

Coefficient 
m-SiA8 

V2 

Coefficient 

c-  /  y8 

A/S^r" 

1.040 
1.387 
1.410 

.001754 
.001255 
.001200 

2.817 
3.139 
3.221 

.00023447 
.00020774 
.000193711 

65.30 
69.40 
71.  85 

REMARK.— Gravelly  bottom  reduces  C   as  depth     decreases. . 
See  §  13. 

Clean  caual  in  firm  earth  and  in  beet  order.    Straight.  (Watt.) 


Surf. 
Wdth 
Feet 

Dpth 
Feet 

R 

Feet 

S 
Slope 

V 

Feet  Sec. 

Coefficient 
m-  S^r3 

Coefficient 
0=   /    v8 

V* 

VST/r* 

18.00 

4.00 

2.40 

.0000631 

1.134 

.00018216 

74.10 

Mill  race.  Flachau,  Hungary.    Clean  ditch  in  firm  earth. 
(Rittinger.) 


0.55 
0.86 

0.467 
0.703 

.0020 
.0020 

1.953 
2.199 

.00016742 
.00024350 

77.46 
64.58 

REMARK. — This  is  an  example  of  remarkably  bad  guaging. 
The  writer  has  never  known  the  smoothest  and  firmest  peri- 
meters of  earth  with  best  alignment  to  develop  a  higher  value 
of  C  than  75.00.  In  a  clean  canal  with  earth  perimeter  there 
should  be  very  little  variation  in  C,  especially  for  BO  small  a 
change  in  depth.  Rittinger's  experimental  data  of  flow  are 
usually  much  better  than  the  average  of  such  data,  but  the 
above  is  evidently  untrustworthy. 

Linth  canal.  Grynau.  Clean  canal  in  common  loam. 
Side  slopes  a  little  irregular.  Rod  floats.  (Legler.) 


Surf. 
Wdtb 
Feet 

Dpth 
Feet 

R 
Feet 

S 
Slope 

V 

Feet  Sec. 

Coefficient 

--*£ 

Coefficient 

^jg 

123.00 

5.14 
7.12 

8.87 
9.18 

.00029 
.00032 
.00036 
.00037 

3.414 
4.418 
5.40 
5.53 

.0002900 
.0003115 
.0003278 
.00033648 

58.74 
56.70 
55.30 
54.52 

REMARK.— The  slight  irregularities    of    the    side  slopes 
cause  the  value  of  C  to  decrease  as  the  depth  increases  and 


SULLIVAN'S  NEW  HYDRAULICS. 


81 


includes  a  greater  proportion  of  side  elope  perimeter  which  is 
rougher  than  the  bottom.  The  results  of  guagings  in  such 
channels  as  this  are  probably  what  led  Kutter  to  suppose 
that  the  value  of  C  would  decrease  with  an  increase  in  slope 
where  RT>1  meter:  The  perimeter  above  the  usual  depth  of 
flow  in  a  channel  of  any  size  whatever  is  exposed  to  freezing 
and  thawing,  the  burrowing  of  insects  and  the  growth  of  veg- 
etation. The  change  of  slope  or  of  hydraulic  radius  has  no 
effect  upon  the  roughness.  The  value  of  C  depends  upon  the 
mean  of  the  different  degrees  of  roughness.  See  NOB.  8,  9,  10, 
Group  No.  8,  and  Solani  Embankment,  Group  No.  6,  where 
the  hydraulic  radii  are  both  less  and  greater  than  one  meter 
or  3.281  feet,  and  where  the  slopes  increase  with  R.  It  will 
be  seen  that  it  is  the  roughness  of  perimeter  alone  that  af- 
fects the  unit  value  of  C  and  that  C  varies  with  f/  r8  only, 
from  its  unit  value  as  tabled  for  the  same  degree  of  rough- 
ness. Kutter's  C  should  vary  only  as  {/r  for  any  given  de- 
gree of  roughness,  and  for  different  degrees,  it  should  vary 
as  the  mean  of  the  roughness  and  as  J/r,  but  should  not  be 

o 

affected  by  the  slope  at  all,  because-^-  is  necessarily  con- 
stant for  all  slopes.  The  recent  gaugings  of  the  Mississippi 
entirely  explode  Kutter's  theory. 

GROUP  No.  10.  RIVERS. 

Mississippi  River,  Carrolton,  La.  Bottom  is  fine  sand 
and  the  sides  of  alluvium,  fairly  stable.  (Miss.  River  Com. 
Report,  1882.) 


Surface 
Width 
Feet 

Depth 
Feet 

R 
Feet 

S 

Slope 

V 

Feet 
Sec. 

Coefficient 
m_Sy/r* 
y 

Coefficient 

0=J     v* 

VSv/r3 

2647.00 
2565.00 
2582.00 
2359.00 
2423.00 

93.00 
90.00 
92.00 
86.00 
89.00 

63.10 
63.40 
57.20 
57.60 
57.70 

.0000165 
.0000127 
!  0000139 
.0000097 

.0000112 

5.90 
5.08 
4.46 
2.95 
3.73 

.00024000 
.00025000 
.00032220 
.00048745 
.00035300 

64.54 
63.25 
58.31 
44.78 
59.72 

REMARK. — The  values  of  R  were  taken  as  nearly  equal  as 
could  be  selected  from  the  Report  so  that  slope  alone  would 
show  its  effect  in  connection  with  the  various  degrees  of  rough- 
ness at  different  depths.  The  writer  acknowledges  that  he 
has  little  confidence  in  the  correctness  of  these  guagings,  but 
the  various  slopes  and  velocities  tabled  probably  bear  some  re- 
lation to  the  actual  slopes  and  velocities.  It  does  not  appear 
that  fhe  value  of  either  Kutter's  C  or  that  of  the  writer  de- 
creases as  slope  increases.  The  values  of  Kutter's  C  for  the 


82 


SULLIVAN'S  NEW  HYDRAULICS. 


above  guagings   are   given   below,   as   transcribed   from  hie 
work. 


Surface 
Width 
Feet 

Depth 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Kutter'e 
C 

Kutter's 
n 

2359.00 
2423.00 
2582.00 
2565.00 
2647.00 

86.00 
89.00 
92.00 
90.00 
93.00 

57.60 
57.70 
57.20 
63.40 
63.10 

.0000097 
.(1000112 
.0000139 
.0000127 
.0000165 

2.95 
3.73 
4.46 
5.08 
5.     £0 

124.8 
146.7 
158.2 
179.0 
182.9 

.0452 
.0354 
.0290 
.0261 
.0218 

REMARK. — The  value  of  the  writer's  C  for  a  depth  of  86 
feet,  in  the  first  table  above,  corresponds  with  the  average 
value  of  C  for  rough,  sandy  perimeters  in  rivers,  and  is  prob- 
ably about  the  true  value  for  this  place.  These  are  double 
float,  or  mid-depth  guagings.  Kutter's  n  is  not  as  constant  or 
as  good  an  index  of  roughness  as  is  chaimed  for  it. 

Sacramento  River,  Freeport,  California.  Bottom  of  shift- 
ing sand.  Sides  of  earth.  Straight  reach.  Guaged  by  me- 
ter. (C.  E.  Grunsky.) 


Date  of 
Guag- 
ings 

Area 
Sq. 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 
n-8^8 

Coefficient 

c=Vs^ 

V* 

March  llrh 
14th 

17th 
18th 
19th 
28th 
May     26th 

14,540 
14,920 
14.880 
14.750 
14.690 
14,570 
12,160 

23.45 
23.99 
23.92 
23.79 
23.69 
23.54 
19.93 

.0000744 
.0000786 
.0000675 
.0000750 
.0000713 
.0000778 
.0000580 

3.994 
4.157 
3.974 
3.897 
3.741 
3.383 
2.879 

.00053000 
.00052867 
.00050000 
.00057290 
.00058720 
.00077637 
.00062246 

43.45 
43.49 
44.72 
41.78 
41.27 
35.89 
40.08 

REMARK.— The  low  water  area  at  this  place  is  4,590  square 
feet.  From  the  dates  and  areas  given  it  will  be  seen  that  the 
guagings  were  made  during  high  water,  and  that  the  river 
was  not  stationary,  or  that  continual  scour  or  fill  was  going 
on.  Mr.  Grunsky  says  in  his  report:  "The  river  bottom  is 
sand.  The  river  is  there  (at  Freeport)  surcharged  with  sand 
brought  in  by  its  tributaries  in  quantities  greater  than  the 
water  can  assort,  according  to  volume  and  yelocitiy  of  flow. 
At  the  high  stages  of  the  river  the  changes  in  the  contours  of 
the  bottom  are  rapid  and  sometimes  sudden.  Boils  are  of  fre- 
quent occurrence.  The  river  is  full  of  whirls  "  (Report,  p. 
86.)  At  pages  96,  97  of  his  report  Mr.  Grunsky  says:  To  pre- 
pare a  scale  of  discharge  representing  the  volume  of  the 
river's  flow  at  various  elevations  of  the  water  surface,  for  a 


SULLIVAN'S  NEW  HYDRAULICS. 


locality  such  as  Freeport,  was,  in  view  of  the  shifting  position 
of  the  river  bottom,  an  uncertain  undertaking.  *  *  * 
Neither  could  any  reasonably  correct  relation  between  water 
surface  elevation  and  velocity  be  established." — Report  Com. 
Public  Works  to  Governor  of  California,  1894. 

River  Rhine  in   Rhine   Forest.     Bed     of  coarse    gravel. 
(La  Nicca.) 


R 
Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 
ro-Sv/r" 

Coefficient 

n-   /    V 

V2 

VSv/r» 

0.42 
0.76 
1.21 

.0142 
.0142 
.0142 

2.332 
4.526 
6.032 

.000710 
.000483 
.000520 

37.5"> 
45.50 
44.47 

Simme   Canal.      Canton     Berme.      Very   coarse    gravel. 
(Wampfler.) 

1.32 
1.36 

1.82 
1.87 

.0170 
.0116 
.0065 
.0070 

5.993 
4.491 
4.92 
5.373 

37.37 

40.38 
38.94 
40.16 

Mississippi    River.    Columbus,    Ky.      Rocky  bluffs   and 
gravel.     (Humphreys  &  Abbott.) 

65.90   |   0000658   I    6.957       .OX)7516         36.48 

Mississippi  River,  Vicksburg,   Miss.      Rocky   bluffs  and 
Gravel.    (Humphreys  &  Abbot. 


.000678 


River  Izar.     Coarse  gravel  bed.     (G.  ebenau.) 


River  Rhine.     Boulders  and  gravel.     Large  stones  on  the 
bottom. 

21.65       .001    I    8.858        .001284         27.92 


84  SULLIVAN'S  NEW  HYDRAULICS. 

River  Seine  at   Paris.    Fairly    regular  reach, 
while  rising.     (Poiree.) 


Guaged 


Area 

R 

S 

V 

Coefficient 

Coefficient 

Sq.   Feet 

Feet 

Slope 

Feet  Sec. 

Sv/r8 

p  _     /     V* 

V8 

V  SvA8 

1978 

5.70 

.000127 

2.094 

.0003940 

50.38 

2570 

7.10 

.000133 

2.264 

.0004909 

45.13 

3176 

8.40 

.000135 

2.418 

.0005620 

42.18 

3692 

9.50 

.000140 

3.370 

52.85 

4421 

10  90 

.000140 

3.741 

!  0003600 

52.71 

5108 

12.20 

.000140 

3.816 

.0004090 

49.44 

6372 

14.50 

.000140 

4.232 

.0004316 

48.12 

6929 

15.00 

.000140 

4.512 

.0004000 

50.00 

8034 

15.90 

.000172 

4.682 

.0004974 

44.84 

8668 

16.80 

.000131 

4.800 

.0003915 

50.54 

9522 

18.40 

.000103 

4.689 

.0003700 

51.98 

REMARK.— The  guagings  were  made  by  floats,  bazin  says 
they  are  good.  It  is  seen  from  the  areas  recorded  that  the 
river  was  rising.  Considering  tb.3  different  degrees  of  rough- 
ness of  the  sides  as  the  water  rose  above  its  usual  depth  of 
flow,  and  the  great  difficulty  of  ascertaining  the  true  slope 
on  a  rapidly  rising  river  the  results  are  quite  satisfactory 
The  slope  for  r=15.90  is  probably  an  error.  See  the  discus- 
sion of  these  and  other  data  by  Gen.  H.  L.  Abbot  in  the 
Journal  of  the  Franklin  Institute  for  May,  June,  July,  1873. 
The  guagings  of  the  Seine  at  Triel,  Menlon  and  Poissy  have 
been  condemned  because  the  water  surface  was  affected  by 
tidal  oscillations  as  great  as  two  feet  while  the  guagings  and 
slopes  were  taken.  It  was  also  rising  at  that  time  as  shown 
by  the  areas.  The  slope  of  the  water  surface  under  such  con- 
ditions could  not  be  determined  with  any  accuracy. 

Ohio  River,  Point  Pleasant,  W.  Va.,  Mid-depth  floats.  (Ellet). 


Area 
Sq.  Feet. 

R 
Feet 

S 

Slope 

v 
Ft  Sec 

Coefficient 
Sr/r3 
m=-V- 

Coefficient 

r      /    v* 

^-AMVr" 

7218.00 

6.72 

.0000933 

2.515 

.000257 

62.38 

Great      Nevka    River,    Surface    floats.    8-10    rule    applied. 
(Destrem). 


15554.00 

17.40  | 

.0000149 

2.049 

.00025748   | 

62.32 

Mississippi    River.     Quincy, 
Clarke). 


111.      Sandy    alluvium.     (T.   C. 


15911.00 
51610.00 

9.87 
16.27 

.00007434 
.00007434 

2.941 

3.898 

.00026526 
.00032135 

61.40 
55.78 

SULLIVAN'S  NEW  HYDRAULS. 


85 


REMARK. — The  first  gauging  at  Quincy  was  at  low  water 
when  the  flow  was  entirely  in  contact  with  its  usual  peri 
meter  which  is  somewhat  smoother  and  less  irregular  than 
the  banks  above  the  usual  lo?7  water  depth.  The  second 
gauging  was  at  high  water  after  permanent  high  water  con- 
ditions had  obtained.  The  slope  of  water  surface  was  the 
same  for  both  stages  of  the  river,  showing  that  stationary 
conditions  had  occurred. 


Speyerbach  Creek.    Firm  earth  bed.                           (Grebenau) 

Area 
Sq.   Feet 

R 
Feet 

S 
Slope 

V 
Feet  Sec. 

Coefficient 

Coefficient 

m=^l. 

Vs 

'       va 

r<        / 

-Vs^Ti- 

30.20 

1.54 

.0004666 

1.814 

.00026931 

60.93 

River  Neva.    Surface  floats.    8  10  rule  applied- 
43461.00  35.40       .000139?   I    3.23    I 


(Destrem). 
59/70 


River  Elbe.      Steep    banks.      Coarse  gravel   and   small 
boulders.     (Harlacher.) 


Surface 
Width 
Feet 

Depth 
Feet 

R 
Feet 

S 
Slope 

V 
Feet 
Sec. 

Coefficient 

S^r" 
m=  —  ^  

V8 

Coefficient 

C=A/s£r 

343  00 

452.00 

6.20 
11.80 

3.51 

7.77 

.00038 
.00041 

2.49 
4.95 

0004030 
.0003708 

49.80 
52.00 

REMARK.— In  a  channel  like  this  with  gravel  and  small 
bouldeis  on  the  bottom  the  value  of  Cfor  a  depth  of  1.50  feet 
would  not  exceed  40  if  the  channel  were  narrow.  In  a  wide 
bottomed  rough  channel  with  steep  banks  smoother  than  the 
bottom,  it  will  require  a  considerable  depth  of  flow  to  include 
sufficient  side  wall  to  balance  the  rougher  bottom  perimeter. 
The  above  guagings  were  by  meter. 

River  Salzach,  Bavaria.  At  different  places  and  stages. 
Meter.  (Reich.) 


R 
Feet 

S 

Slope 

V 
Feet 
Sec. 

Coefficient 
m=^r* 

V2 

Coefficient 

C=A/S7FT 

3.45 
3.52 
4.96 
5.00 
5.20 
7.00 

.000280 
.000348 
.000290 
.000607 
.000410 
.00036 

2.686 
3.618 
3.510 
5.543 
5.094 
4.118 

.0002487 
.0001760 
.0002600 
.0002190 
.0001800 
.000393 

63.40 
75.40 
62.00 
67.60 
74.53 
50.44 

SULLIVAN'S  NEW  HYDRAULICS. 


REMARK.— These  guagings  were  made  in  1885  with  the 
meters  then  in  use.  The  nature  of  the  perimeter  is  not  stated, 
but  it  is  safe  to  state  that  no  natural  channel  will  develop  a 
value  of  C  as  high  as  75.  The  mill  race  at  Pricbratu  with  its 
masonry  side  walls  and  smooth  clay  bottom,  and  the  smooth- 
est, best  aligned  canals  in  firm  earth  and  in  perfect  order,  only 
give  C=T5.  See  next  group. 

GROUP  No.  11— CANALS. 

Mill  race  at  Pricbram,  Hungary.  Side  walls  of  masonry. 
Smooth  clay  bottom  1.88  feet  width.  Trapezoidal.  (Rittin- 
ger.) 


Depth 
Feet 

R 
Feet 

S 
Slope 

V 

Feet  Sec. 

Coefficient 
m=Ji^ 

Vs 

Coefficient 

n_   /    V 

A/S;/r» 

0.54 
0.66 

0.373 
0.425 

.0010 
.0010 

1.127 
1.254 

.00017937 
.00017621 

74.67 
75.33 

See    Nos.     11,    12     and   13  in  group    No.    8,    also    see 
group  No.  9. 

Realtore  canal.    Common  loam  bed   in   only   fair   condi- 
tion. (D'Arcy  &  Bazin.) 


Surface 
Wdth 
Feet. 

Dpth 
Feet 

R 
Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient  1  Coefficient 

m_S;/r«     |c        /     v* 

v»                 V  Sv/r* 

19.70 

4.50 

287 

.00043 

2.54    !       .000324667      !           55.51 

Marseilles  canal.     Common  loam  bed  in  only  fair   condi 
tion.        (D'Arcy  &  Bazin.) 


2.90     I  .00043     2.536 


Henares  canal,  Spain.     Common  loam  bed   in  fair  condi- 
tion.   (See  Fanning.) 


4.92    |    2.95    |. 000326  |  2.2%    |       .000313328 


Lauter  canal.    Gravelly    soil.     Bed    in    fair    condition. 
(Strauss.) 


29.50 

1.82 

.000664 

2  106 

.00036758 

52.16 

See  Group  No.  8,  California  canals. 

Rivers,  creeks  and  canals  grouped  according  to  roughness 


SULLIVAN'S  NEW  HYDRAULICS. 


87 


of  perimeter  at  the  given  depths.  Areas  are  given  in  all  cases 
where  known.  Perimeters  are  described  as  fully  as  available 
information  will  permit.  The  guagings  are  good,  bad  and 
worthless.  It  is  difficult  to  separate  them  without  more  pre- 
cise knowledge  of  the  exact  conditions  under  which  they  were 
made.  The  slopes  of  some  rivers  were  measured  while  the 
stream  was  rising  and  the  velocities  were  taken  when  the 
stream  was  falling.  A  fair  average  value  of  C  may  be  arrived 
at  for  each  class  of  perimeters  from  what  has  already  been 
shown  together  with  the  following  groups. 

GROUP  No.  12. 

Shallow  canals  grown  up  in  weeds  reaching  nearly  to  the 
water  surface. 


Name    of 
Channel 

Area  I     R 
J^etJFeet 

S 

Slope 

V 

Feet 
Sec. 

Coefficient 
Sv'r* 

m  =  .-*- 
Vs 

Coefficient 

°=V^ 

Visalia 
Canal 
Viealia 
Canal 

!    1.77 
j     1.13 

.00035 
.00060 

0.845 
0.868 

.001154 
.00095658 

30.00 
32.33 

GROUP  No.  13. 

Large  canals  with  quantities  of  weeds  and  bushes  on  the 
margins  and  shallow  places. 


Cavour 
Canal 
C.  &0.  C. 
Feeder 
C.&O.  C. 
Feeder 

799.10 
119.00 
121.00 

5.58 
3.70 
3.70 

.000357 
.0006985 
.0006985 

2.60 
2.723 
3.032 

.0006964 
.0006700 
.0005410 

37.85 
38.64 
43.00 

GROUP  No.  14. 

Large  streams  with   very  rough   banks   and   with   large 
stones  and  gravel  on  the  bottom . 


River 
Rhine 
River 
Rnine 
River  Izar 

13725.50 

4650.10 
1063.40 

21.65 

7.67 
6.04 

.00100 

.00125 
.00250 

8.858 

4.921 
7.212 

.001284 

.001096 
.000710 

27.92 

30.20 
37.54 

88  SULLIVAN'S  NEW  HYDRAULICS. 

GROUP  No.  15. 

Shallow   channels  with   very    coarse    gravel    and   email 
boulders  on  the  bottom. 


Name   of 
Channel 

Dpth 

Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 

Vs 

Coefficient 

c      '   v 

\  fcn/rs 

Schwarza 
River 

0.95 

0.80 

.0090 

2.528 

.001007 

31.47 

Schwarza 

River 

1.20 

0.99 

.0052 

2.544 

.0007914 

35.55 

Solani  Em- 

bankment 

1.50 

1.69 

.000090 

0.440 

.00102128 

31.28 

Bimme 

Canal 

1.32 

.0170 

5.993 

.00072 

37.37 

River 
Rhine 

0.42 

.0142 

2.332 

.00071 

37.50 

GROUP  No.  16. 

Channels  with  one  rough,   stony  bank  and  with   gravel 
bottoms.     One  bank  of  earth. 


Name   of 
Channel 

Dpth 

Feet 

R 

Feet 

S 

Slope 

V 

Feet 

Coefficient 

V2 

Coefficient 

M'ssissippi 

River* 

88.00 

65.90 

.0000680 

6.957 

.0007516 

36.48 

M  ssissippi 

Riverf 

100.00 

64.10 

.0000638 

6.949 

.0006780 

38.40 

Grosbois 

CanalJ 

1.70 

1.05 

.0039360 

1.080 

.0008720 

33.86 

Grosbois 

Canal 

2.30 

1.37 

.0009360 

1.370 

.0007997 

35.36 

Grosbois 

Canal 

2.60 

1.52 

.000957 

1.560 

.0007310 

36.98 

Grosbois 

Canal 

2.90 

1.64 

.000964 

1.710 

.0006923 

38.00 

*At  Columbus,  Kentucky.  Blutf  on  left  bank  composed 
of  strata  of  coarse  sand,  coarse  brown  clay,  blue  clay,  fine 
sand,  coarse  gravel,  limestone,  pudding  stone,  iron  ore. 

f  At  Vicksburg,  Miss.  Bluff  forms  left  bank  and  is  com- 
posed of  strata  of  blue  clay,  logs,  carbonized  wood,  marine 
shells,  sand  full  of  shells,  sandstone.  See  "Levees  of  the  Mis- 
sissippi River,"  by  Humphreys  &  Abbot,  pages  28,29.  (1867.) 

JGravel  and  pebbles  on  the  bottom.  One  side  slope  rip- 
rapped  with  rough  stone— the  other  side  slope  of  earth. 


SULLIVAN'S  NEW  HYDRAULICS.  89 

GROUP  No.  17. 

Channels  in  firm  earth  with  low  stumps  and  roots  on  the 
bottom. 


Name   of 
Channel 

Area 

X, 

R 

Feet 

s 

Slope 

V 

Feet 
Sec. 

Coefficient 

-*£ 

Coefficient 
•                     . 

A/fcVr" 

Bayou  Pla- 
quemine 
Bayou  Pla- 
quemine 

3560.00 
4259.00 

18.30 
15.30 

.0002064 
.0001437 

5.198 
3.959 

.000598 
.00054875 

40.88 
42.69 

REMARK. — This  bayou  was  guaged  by  Humphreys  &  Ab- 
bot with  mid-depth  floats.  It  is  simply  an  overflowed  coule, 
which  was  formerly  covered  by  a  thick  forest  of  cypress 
trees.  These  trees  were  cut  down  and  the  water  brought 
into  the  Plaquemine  in  1770  by  means  of  a  small  canal  con- 
necting with  the  Mississippi  river.  As  the  dirt  washed  from 
around  the  stumps  the  Navigation  company  had  them  recut. 
See  "Levees  of  the  Mississippi  River,"  page  204,  note.  This 
bayou  varies  in  width  from  200  to  300  feet,  and  in  depth  from 
20  to  35  feet.  There  is  luxuriant  plant  growth  along  the  mar- 
gins. 

GROUP  No.  18. 

Grosbois  canal.  Earth  bed  in  bad  repair,  with  many 
patches  of  vegetation. 


Surface 
Width 
Feet 

Dpth 
Feet 

R 

Feet 

S 

Slope 

Feet 
Sec. 

Coefficient 
m-  S^r« 

Vs 

Coefficient 

°-J& 

10.10 
12.30 
13.50 
14.70 

1.70 
2  40 
2.80 
3  10 

1.06 
1.41 
1.60 
1.76 

.00042 
.00047 
.00047 
.00045 

0.89 
1.18 
1.31 
1.39 

.0005785 
.000u6505 
.000554326 
.000o44 

40  95 
42.06 
42  47 

42.86 

90 


SULLIVAN'S  NEW  HYDRAULICS. 


GROUP  No,  19. 
Channels  with  gravelly  bottoms  and  rough,  irregular    banks 


Name  of 
Channel 

Area 
Sq. 
Feet 

R 
Feet 

S 
Slope 

Feet 
Sec. 

Coefficient 
Sv/r" 

Coefficient 

m  —       V8 

Mississippi 
River 
Feeder 
Chazilly 
Feeder 
Chazilly 
River  Rhine 
Seine  (Poissy) 

150365.00 
11.30 

18.80 
19135.00 
10400.00 

57.40 
1.04 

1.41 

16  50 
17.80 

.0000481 
.0004450 
.0009930 

.(HXHIWT 
.0000750 

6.310 
0.962 

1.789 
3.575 
3.330 

.00052400 
.00053100 

.00051450 
.00051235 
.00050800 

43.69      • 
43.40 

44.10 
44.17 
44.37 

GROUP  No.  20. 

Rivers  and  Canals  with  beds  of  sand  and  with   irregular  side 
Slopes  of   earth. 


Name  of 
Channel 

Area 
Sq. 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec 

Coefficient 

Coefficient 

V2 

Seine 

8034.00 

15.90 

000172 

4.682 

.0004974 

44.83 

^Pari.) 

2570.  00 

7.10 

000133 

2.264 

.0004909 

45.13 

Chazilly 

22.20 

1.54 

000986 

1.959 

.00049 

45.16 

Feeder 

Chazilly 

9.50 

0.96 

000792 

1.234 

.0004878 

45.28 

Feeder 

Grosbois 

23.00 

1.63 

000479 

1.434 

.00048456 

45.40 

Mississip- 

pi River 
Seine 

179502.00 

64.50 

.0000436 

6.825 

.0004845 

45.42 

(Poissy) 

9733.00 

16.80 

.0000670 

3.101 

.00048 

45.65 

Seine 

(Triel) 
Seine 

6375.00 

12.40 

.0000600 

2.359 

.0  047 

46.12 

(Poissy) 

8996.00 

15.90 

.0000620 

2.911 

.0004638 

46.43 

Feeder 

Chazilly 

•     22.90 

1.57 

.0004350 

1.401 

.000456 

46.83 

M?s^6River 

5010.10 

10.85 

.0000289 

1.509 

.0004536 

46.95 

Saalach 

96.76 

1.34 

.001164011.970 

.00045238 

47.02 

Seine 

(Poisay) 

7475.00 

13.60 

.00005002.372 

.0004457 

47.34 

See  Sacramento  River,  Group  No,  10. 


SULLIVAN'S  NEW  HYDRAULICS,  91 

GROUP  No.  21. 

Natural  Channels  with  sandy  gravel  bottoms  and  fairly 
regular  sides  of  earth.    Canals  in  common  loam  in  bad  repcir 


Name  of 

Area 

R 

S 

V 

Coefficient 

Coefficient 

§q' 

Feet 

Sv/r3 

c     /  v* 

Channel 

Feet 

Feet 

Slope 

Sec 

va 

—  \  a    

Feeder 

Grosbois 

19.40 

1.41 

.000858 

1.815 

.00043515 

47.90 

Feeder 

Grosbois 

18.10 

1.38 

.00045 

1.296 

.00043280 

48.07 

Feeder 

Grosbois 

27.20 

1.71 

.000441 

1.510 

.00043230 

48.08 

Seine 

(Paris) 

6372.00 

14.50 

.00014 

4.232 

.0  04316 

48.12 

Feeder 

Grosbois 

25.40 

1.69 

.00033 

1.296 

t.  00043155 

48.14 

Seine 

(Poissy. 

7952.00 

14.20 

.000054 

2.595 

.00042900 

48.28 

Izar 

300.10 

1.85 

.0025 

3.997 

.00042430 

48.54 

Feeder 

Grosbois 

17.20 

1.38 

.00045 

1.326 

.000414 

49.18 

Feeder 

Grosbois 

20.90 

1.56 

.000525 

1.575 

.0004138 

49.23 

Feeder 

Grosbois 

32.00 

1.85 

.00  '33 

1.411 

.0004168 

49.00 

Feeder 

Grosbois 

22.90 

1.56 

.000842 

1.998 

.00041 

49.39 

Feeder 

Chazilly 

14.10 

1.18 

.000929 

1.703 

.00041068 

49.34 

Seine 

(Paris) 

5108.  CO 

12.20 

.00014 

3.816 

.0004  9 

49.44 

Visalia 

Canal 

1.16 

.00175 

2.33 

.0004026 

49.83 

Rhine 

14149.80 

9.72 

.OOJ112 

2.91 

.0004008 

49.95 

Seine 

(Paris 

6929.00 

15.00 

.00014 

4.512 

.0004 

50.00 

Seine 

(Paris) 

1978.00 

5.70 

.000127 

2.094 

.000394 

50.38 

Salzacb 

40  5.60 

7.00 

.00036 

4.118 

.  00393 

50.44 

Seine 
(Paris) 

8668.00 

W.80 

.000131 

4.80 

.0003915 

50.54 

Feeder 

Grosbois 

26.80 

1.71 

.000493 

1.683 

.00039 

50.64 

Feeder 

Grosbois 

14.90 

1.21 

.000808 

1.667 

.000387 

50.83 

Feeder 

Grosbois 

25.90 

1.71 

.000515 

1.746 

.0003775 

51.46 

Feeder 

Grosbois 

15.40 

1.30 

.000555 

1.480 

.00037557 

51.60 

Seine 

(Paris) 

9522.00 

18.40 

.(XWlCtt 

4.689 

.00037 

51.98 

Lauter 

Canal 

56.40 

1.82 

.000664 

2.106 

.00036756 

52  16 

Saalach 

86.90 

1.38 

.0010357 

2.155 

.000362 

52.56 

Seine 
(Paris) 
Seine 

3692.00 

9.50 

.00014 

3.37 

.0003578 

52.85 

(Paris) 

4421.00 

10.90  j.  00014 

3.741 

.00036 

52.71 

92  SULLIVAN'S  NEW  HYDRAULICS. 

GROUP  No.  22. 

Channels  revetted  with  rough  angular  rubble,  dry  laid; 
channels  in  firm  earth  with  rough,  uneven  bottoms  and  irreg- 
ular side  slopes. 


Name    of 
Channel 

Area 
Square 
Feet 

R 

Feet 

s 

Slope 

V 

Feet 
Sec. 

Coefficient 

m-Syr° 

Coefficient 

C=    /     V 

V8 

VSv/r* 

River 

Waal* 

14782.00 

11.10 

.0001044 

3.165 

.00038607 

50.88 

Turlock 

Rock  Cut 

5.90 

.0015 

7.50 

.000382 

51.18 

Mississippi 

River 

134942.00 

52.10 

.0000303 

5.558 

.0003693 

52.03 

Mississippi 
River 

193968.00 

72.00 

.0000205 

5.929 

.000356 

53.00 

Bayou  La 

Fourche 

286'.  00 

15.70 

.0000438 

2.789 

.000352 

53.30 

Bear  River 

Canal 

5.63 

.00018939 

2.67 

.00035487 

53.00 

River 

Rhine 

6304.00 

11.20 

.0000999 

3.277 

.0003486 

53.57 

Seine 

LMeulan] 

6488.00 

7.70 

.000087 

2.313 

.00034734 

53.66 

*See  Group  No.  7. 

GROUP  No.  23. 

New  canals  in  loam,  or  light  soil,  just  completed;  old  can- 
als in  similar  soil  with  weeds  along  the  margin;  canals  in 
fairly  go  3d  condition  but  with  pebbles  on  the  bottom. 


Name    of 
Channel 

Area 
Square 
Feet 

R 

Feet 

S 
Slope 

V 

Feet 
Sec. 

Coefficient 

Coefficient 
n_   pH 

m         v» 

VsT/r"~ 

Visalia 
Canal 

Visalia 

Idaho0!?81 
&I  Canal 
Feeder' 
Grosbois 

11.80 

0.85 
0.99 
7.232 
1.050 

.0010 
.'006 
.00037878 
.00031 

1.52 
1.33 
4.70 
0.817 

.0003392 
.000334 
.00033347 
.000343 

54.30 
54.72 
54.77 
54.03 

REMARK— The  flow  in  a  new  canal  is  never  as  great  at 
first  as  it  will  become  after  the  bed  is  saturated  with  water 
and  the  loose  material  is  dissolved  and  deposited  in  the  pores 
of  the  earth  and  in  the  little  depressions  and  irregularities 
along  the  perimeter. 


SULLIVAN'S  NEW  HYDRAULICS, 


93 


GROUP  No.  24. 

Clean  channels  with  bottoms  of  fine  gravel  and  sand 
well  settled,  and  with  banks  of  sandy  loam;  channels  in 
sandy  alluvium;  canals  in  ordinary  loam  in  fair  condition  but 
not  recently  cleaned.  (Average  for  canals  in  ordinary  con 
dition). 


Name   of 
Channel 

Area 
Sq. 
Feet 

R 

Feet 

S 
Slope 

Feet 
Sec. 

Coefficient. 
Sj/r3 

•  Coefficient 

m—     vs 

Marseilles 
Canal 

66.00 

2.90 

.00043 

2.536 

.00033 

55.04 

Eealtore 

Canal 

2.87 

.00043 

2.540 

.000324667 

55.51 

Eiver 

Tiber 

2355.00 

9.40 

.001306 

3.413 

.00032335 

55.62 

Seine 

(Meulan) 

5982.00 

7.10 

.00009 

2.31 

.000318876 

55.75 

Eiver 
Ha  me 

306.40 

5.70 

.0001559 

2.558 

.0003213 

55.71 

Miss.  River 

51610.00 

16.27 

.00007434 

3.898 

.00032135 

55.78 

Henares 

Canal 

2.95 

.000326 

2.296 

.000313328 

56.40 

Mississip- 

pi Eiver 

78828.00 

31.20 

.0000223 

3.523 

.0003132 

56.49 

Visalia 

Canal 

1.40 

.00302 

4.02 

.00030951 

56.87 

Feeder 

Grosbois 

18.00 

1.42 

.00029 

1.26 

.000309 

56.89 

*  At  Quincy,  Illinois,  gauged  by  Thomas  C.  Clarke,  C.  E. 
GROUP  No.  25. 

Rivers  and  canals  in  good  condition,  having  beds  of  fine 
sand  and  small  pebbles,  with  fairly  regular  banks  of  firm 
loam. 


Name  of 

Area 

R 

S 

V 

Coefficient 

Coefficient 

Sq. 

Feet 

S;/  r8 

C/ 

Channel 

Feet 

Feet 

Slope 

Sec. 

V2 

Hooken- 

bach  Creek 
Hocken- 

10.30 

0.88 

.0007966 

1.463 

.000307287 

57.00 

bach  Creek 

10.50 

0.87 

.0007783 

1.440 

.00030463 

57.28 

Misssissip- 
pi  Eiver 

195349.00 

72.40 

.0000171 

5.887 

.000304 

57.35 

Upper* 

Miss.  Eiver 

3441.00 

4.42 

.0002227 

2.611 

.00030355 

57.40 

Feeder 

Grosbois 

30.80 

1.78 

.000275 

1.467 

.0003034 

57.48 

Feeder 

Grosbois 

10.90 

0.96 

.00025 

0.886 

.0003 

57.74 

SULLIVAN'S  NEW  HYDRAULICS. 


*  At  Fort  Snelling,  Minnesota.  It  is  apparent  from  the 
areas  and  hydraulic  mean  radii  in  Group  25,  that  the  effect 
of  rough  or  smooth  perimeter  is  the  same  in  a  very  large 
channel  as  in  a  very  small  one.  These  perimeters  are  almost 
exactly  alike,  and  develop  like  coefficients,  regardless  of  size 
or  slope. 

GROUP  No.  26. 

Rivers  and  canals  in  alluvial  soil,  or  firm  earth  mixed  with 
tine  eand,  in  good  condition,  and  free  of  stones  and  weeds. 


Name    of 

Area 

R 

S 

V 

Coefficient 

Coefficient 

Channel 

Square 
Feet 

Feet 

Slope 

Feet 
Sec. 

m=*v;' 

c=Vs^ 

Bayou  La 

Fourche 

3738.00 

15.70 

.0000447 

3.076 

.0002948 

58.24 

Visalia 

Canal 

0.93 

.(004 

1.110 

.000291 

58.62 

Eiver 

Rhine 

5341.00 

7.60 

.0001174 

2.917 

.00028905 

58.83 

Huben- 

graben 

3.80 

0.59 

.0013 

1.424 

.00029 

58.74 

River 

Neva 

43461.00 

35.40 

.0000139 

3.230 

.00028065 

59.70 

Marseilles 

Canal 

3.386 

.000333 

2.720 

.00028065 

59.70 

GROUP  No.  27. 

Canals  in  heavy  loam  in  excellent  repair;  natural  chan 
nels  with  very  fine  sand  on  firm  and  regular  bottoms  with 
sandy  loam  banks  in  excellent  condition,  free  of  weeds  and 
stones. 


Name    of 
Channel 

Area 
Square 
Feet 

R 

Feet 

S 

Slope 

V     j 
Feet! 
Sec.j 

Coefficient 
B_Sv/r« 

Coefficient 

v8 

Vs 

\Si/r3 

Speyer- 
bach  Creek 

30.20 

1.54 

.0004666 

1.814! 

.00026931 

60.93 

15911.00 

9.87 

.00007434 

2.941; 

.00026526 

61.40 

Feeder 

Grosbois 

24.20 

1.57 

.000246 

1.362! 

.00026080 

61.88 

Great 

Nevka 

15554.00 

17  40 

.0000149 

2.049! 

.00025748 

62.32 

Ohio 

River 

7218.00 

672 

.0000933 

2.515! 

.00025700 

62.38 

SULLIVAN'S  NEW  HYDRAULICS.  95 

GROUP  No.  28. 
Canals  in  smooth  clay  with  loose  pebbles  on  the  bottom. 


Name    of 

Area 
Square 

R 

VPont 

s 

CJIona 

V 

Feet 

Coefficient 
m-8^" 

Coefficient 

c     /  v" 

Feet 

Sec. 

V* 

AWr8 

Visalia* 
Canal 
Feeder 
Grosbois 
Viealia 
Canal 

17.10 

1.32 
1.32 
1.34 

.00194 

.000275 
.00177 

3.510 
1.336 

3.460 

.00023862 
.00023366 
000229-254 

64.73 
65.37 
66.04 

*A  few  weeds  along  the  margin  in  patches. 

GROUP  No.  29. 

Canals  in  very  firm,  heavy  soil,  with  clay  bottoms  worn 
smooth,  but  not  recently  trimmed  and  punned;  natural 
streams  of  good  alignment  with  clay  bottoms,  and  fine  grain- 
ed,  firm  and  uniform  alluvial  banks,  free  of  stones  and  vege- 
tation . 


Name  of 

Area 

R 

S 

V 

Coefficient 

Coefficient 

Sq- 

Feet 

m—   Sv/r8 

IVV* 

Channel 

Feet 

Feet 

Slope 

Sec. 

VB 

~^Sv/r8 

Visalia 

Canal 
Visalia 

2.00 

.000400 

2.255 

.00022249 

67.08 

Canal 
Yssel 

3.35 

.000010 

0.531 

.0002174 

67.82 

T,o,hEiVer 

1930.00 

15.90 

.0001166 

2.773 

.00021728 

67.84 

Katrine 

2.525 

.0001578 

1.7126 

.0002158 

68  05 

Visalia 
Canal 
Bayou  La 

3.34 

.0000375 

1.032 

.00021486 

68.22 

Fourche 
Bayou  La 

3025.00 

13.00 

.0000373 

2.843 

.0002145 

68.27 

Fourche 

2957.00 

12.80 

.0000366 

2.807 

.0002126 

68.53 

REMARK— The  bottom  of  this  portion  of  Bayou  La  Four- 
che is  clay,  and  the  banks  are  leveed.  The  banks  are  of 
heavy,  alluvial  soil.  Its  bends  are  few  and  gentle.  There 
are  no  boils,  whirls,  nor  eddies.  It  resembles  an  artificial 
channel  very  much.  For  a  general  description  see  "Levees  of 
the  Mississippi  River",  page  198. 


96  SULLIVAN'S  NEW  HYDRAULICS. 

GROUP  No.  30. 

Canals  in  very  firm,  smooth,  dense  earth,  recently  cleaned, 
trimmed  and  punned,  and  put  in  the  best  condition. 


Name   of 
Channel 

Area 
Sq. 
Feet 

R 

Feet 

s 

Slope 

V 

Feet 
Sec. 

Coefficient 

Coefficient 

Vs 

^"""VSv/r8 

Visalia 
Canal 
Visalia 
Canal 
Visalia 
Canal 
English 
Canal 

50.00 

1.13 
1.09 
0.92 
2.40 

.00060 
.00060 
.001165 
.0000631 

1.88 
1.87 
2.36 
1.134 

.00020396 
.000195 
.000184553 
.00018216 

70.02 
71.61 
73.73 
74.10 

IS— Roughness  of  Perimeter  Defined.— The  foregoing 
tables  of  coefficients  might  be  greatly  enlarged  by  the  ad- 
dition thereto  of  the  data  of  many  other  pipes  and  channels, 
but  such  matter  would  be  simply  cumulative.  It  is  believed 
that  the  tables  given  cover  all  cases  as  accurately  as  the  pub- 
lished data  will  permit,  and  it  was  not  deemed  neccessary  to 
give  but  a  few  examples  of  each  class  in  order  to  assist  in 
selecting  the  value  of  the  coefficient  in  any  ordinary  case. 
The  inaccuracies  which  abound  in  the  data  of  flow  in  all 
classes  of  pipes  and  channels  are  due  in  great  part  to  the 
failure  of  weir  and  orifice  coefficients.  The  writer  is  aware 
that  a  general  belief  in  the  accuracy  of  weir  measurement  has 
become  very  great,  but  the  fact  remains  that  such  measure- 
ments are  very  frequently  erroneous  to  a  very  considerable 
extent.  Meter  measurements  of  velocity  are  still  less  reliable 
When  better  methods  are  discovered  and  adopted,  we  shall 
have  more  reliable  data  than  we  now  have.  In  the  consid- 
eration of  the  degree  of  roughness  of  any  given  channel,  the 
alignment,  uniformity  of  cross-section,  freedom  from  grit 
gravel,  stones  and  vegetation,  are  not,  by  any  means,  all  that 
are  to  be  considered.  The  nature  of  the  material  in  contact 
with  the  flow,  as  to  density  and  compactness,  is  as  important 
as  any  or  all  other  features.  The  coefficients  show  that  for 
clean  canals  in  earth,  the  value  varies  from  about  56  to  75  as 
the  nature  of  the  earth  forming  the  perimeter  varies.  The 


SULLIVAN'S  NEW  HYDRAULICS.  97 

amount  of  sand,  and  whether  coarse  or  fine,  which  enters  in- 
to the  majority  of  different  classes  of  earth,  hae  a  great  effect 
upon  the  flow  and  upon  the  value  of  the  coefficient.  Every 
table  of  data  of  open  channels  abundantly  proves  the  incor- 
rectness of  the  idea  that  the  character  of  the  perimeter  has 
no  influence  upon  flow  in  very  large  channels.  The  Missis- 
sippi river  at  Columbus  and  Vicksburg  with  depths  of  88  and 
100  feet  respectively,  develop  the  same  value  of  the  coeffi- 
cient as  very  small  ditches  having  the  same  kind  of  perime- 
ter. (See  Group  No.  16).  There  is  no  reason  why  this  should 
not  be  the  case,  and  it  would  be  strange  if  it  were  not  the 
the  case.  The  flow  in  large  rivers  is  nearly  always  overesti- 
mated, especially  where  meters  or  surface  floats  are  used  for 
determining  the  velocity.  Insufficient  attention  has  been 
given  the  character  of  the  perimeter  and  its  effect  upon  the 
flow  of  the  water  in  contact  therewith  and  affected  by  the  re- 
actions therefrom.  The  velocity  of  the  film  of  water  in  con- 
tact with  and  affected  by  the  sides  and  bottom  has  never 
been  considered  of  great  importance  in  determining  the  mean 
velocity  of  the  whole  cross  section  in  large  streams,  and  yet 
if  this  outer  layer  of  water  thus  affected  were  deducted  from 
the  whole,  at  least  one  fourth  the  total  area  would  be  sub- 
tracted. 


CHAPTER  III, 


Of  the   Deduction  of  the  General  Formulas. 


16—  Formulas  in  Terms  of  Diameter  in  Feet.—  For  large 
pipes  and  circular  channels  flowing  full,  a  set  of  formulas  in 
terms  of  diameter  in  feet  will  be  most  convenient.  For 
small  pipes  the  coefficients  may  be  in  terms  of  diameter  in 
inches. 

FORMULA  FOB  Loss  OP  HEAD    BY    FRICTION. 

By  formula  (10)  §6,  the  coefficient  of  friction  n,  is 

S" 

(10) 


Hy  transposition  in  (10)  we  have  the  formula   for   loss   of 
head  in  feet  by  friction 


_  n 

h  ==        ~        lv 


In  which,  h"=  total  head  in  feet  lost  in  the  length   in   feet,  I 
d=diameter  of  pipe  in  feet. 
n=coefficient  in  terms  of  diameter  in  feet. 
Z=length  of  pipe  in  feet. 
v=velocity  in  feet  per  second. 

FORMULA  FOR  HEAD   IN  FEET  LOST  PER  FOOT  LENGTH   OF 

PIPE. 
S" 
As  n  =^jf-X   i/d8,  we  have  by  transposition, 

S"=VcT*Xv*....^  ............................  (22) 

In  which, 

S"=head  in  feet  lost  by  friction  per  foot  length  of  pipe. 
n=unit  value  of  the  coefficient  of  resistance   which  in- 
creases as  vs,  and  is  inversely  as  i/ds. 

FORMULA  FOR  MEAN  VELOCITY  OF  FLOW. 
By  equation  (12)  the  coefficient  of  velocity  m,  is 


SULLIVAN'S  NEW  HYDRAULICS. 
Hd^d S          ff_  S^/d8 

And  by  transposition  in  (12)  we  have, 


In  which, 

v  —  mean  velocity  in  feet  per  second. 

H=total  head  in  feet,  where  discharge  is  free. 

fl=h"+bv  where  discharge  is  throttled  (§  5). 

1=  length  of  pipe  in  feet. 

d=diameter  in  feet. 

m—  coefficient  of  velocity  determined  in  terms  of  d  in 
feet. 

Where  the  altitude  is  sufficient  to  affect  the  value  of  g, 
the  formula  may  be  written, 


,  when  m-      /  y8      ;  H:= 

In  this  case  the  value  of  m  must  be  found  according  to 
the  value  of  2gH  at  the  given  altitude. 

FORMULA  FOR  TOTAL   HEAD   REQUIRED  TO  GENERATE  A 

GIVEN  VELOCITY. 
miv*    mlv* 


The  slope  (S)  required  to  generate  a  given    mean  velocity 
is 

S  =  m-^n  ^-^rrXd* (25) 


To  find  the  length  in  feet  I,  in  which  there  must  be  the 
given  head  in  feet  H,  or  fall  in  feet  equal  H,  in  order  to 
generate  the  given  mean  velocity  v:— 

Hdy/d_Hyyd*  (26) 

m  vs        vs      m 


100  SULLIVAN'S  NEW  HYDRAULICS. 

FORMULAS  IN  TERMS  OP   CUBIC   FEET  PER  SECOND  AND  DI- 

AMETER IN   FEET. 

Letq=cubic  feet  per  second  discharged,=AreaXvelocityi 
a=  area  of  pipe  in  square  feet. 


Then  a=dsX-7584,  and  v=/    X~    whence 


q  v/T~m=d2  .785VH  ^d8    or  q  1/m=dir.7854v/S1/d 

whence 

8 


(28) 

I  .61685  IS    i/d*i_     /  .6165      y  /«a  /HII  . .  .(29) 
9—  \ m  *       m 

Hence  the  diameter  in  feet    required  to   cause   the  dis- 
charge of  a  given  number  of  cubic  feet  per  second  is 

(28) 
If  total  loss  of  head  is  predetermined  then 


And  the  slope  required  to  cause  a  given  diameter  to  dis- 
charge a  given  quantity  in  cubic  feet  per  second  will  be 
m  q*  m  q* 

And, 

q»m?         ._JE_xx__il 
[=  61685^/c 


SULLIVAN'S  NEW  HYDRAULICS.  101 

_° ,    q*    v  i      . .  /am 


.616853/d"    -  .61( 
hVd"  .61685 


H  .GieSSy'd1! 
m  = ; — s : 


_H_JP1685j/dij_ 
m  q* 


/Hyd".61685_     /S|/d"X. 61685  (36) 

I/  "       m  {  K  "  m 

As  v=-^-  it  appears  from  an  inspection  of  (29)  and  (36) 

that  the  relative  discharges  will  be  ao  f/d11. 

The  slopes,  or  heads  and  lengths  being  equal,  then 

q  :  q  :  :  ^/d11  :     fc/d11,    provided     the    roughnesses    are 
equal. 

(See  Table  No.  18,  §  33.) 

17  Formulas  in  Terms  of  Pressure,  Diameter  and 
Quantity. — Head  in  feet  and  pressure  in  pounds  per  square 
inch  are  convertible  terms.  Pressure  increases  directly 
as  head  increases,  and  the  velocity  will  be  proportional  to 
either  ,/H  or  /P.  When  H=2.3041  feet,  P=l  Ib.  per  square 
inch.  Hence  the  coefficients  determined  in  terms  of  H  will 
not  apply  in  a  formula  in  terms  of  P.  The  coefficients  may, 
however,  be  converted  from  terms  of  H  or  S  to  terms  of  P  as 
pointed  out  in  §  10  and  as  follows:-- 
p 

P=HX-434,  and  H=-jg|-=PX23041.  Hence  if  we   have 

the  value  of  n  or  m  in  terms  of  H  and  d  in  feet,  and  wish  to  con- 
vert to  terms  of  pressure  in  pounds  per  square  inch  P,  and 
diameter  in  feet  d,  we  divide  the  value  of  n  or  m  in  terms  of 
H  and  d,  by  2.3041,  and  the  result  is  the  value  of  n  or  m  in 
terms  of  Pand  d. 


102  SULLIVAN'S  NEW  HYDRAULICS. 

Tt,_  n  Zv*         nlv* 


.(38) 


V/d8    '"    dj/d 
In  which, 

P'=total  Ibo.  per  squan  inch  pressure  lost  by  friction. 
n=coefficient  of  resistance  in  terms  of  P'  and  d. 
Let  P=total  pressure  in  Ibs.  per  square  inch. 
P'=total  pressure  lost  by  friction. 
Pv=velocity  pressure. 
When  the  discharge  is  free,  P  =P— Pv, 
To  find  the  pressure  in  pounds  per  square   inch   required 
to  balance  the  friction   and  generate  a  mean   velocity  v,  in 
feet  per  second: — 


«*> 

q=av=$/d11  \/p  X  -7854  -H  ^ 

q^nTT^Kd'VPX  .7854,whence 

To  find  the  diameter  in  feet  to  discharge  the  quantity  q, 
in  cubic  feet  per  second,  through  the  length  I,  when  the 
total  pressure  is  P:— 


>8q*       11  /    m8  ll/Z8q* 

d*=  K -3S66-  X    I/pi"'  "  (43) 

To  find  the  total  pressure  required  to  balance  the  re- 
sistance and  force  the  discharge  of  q  cubic  feet  per  second 
through  a  pipe  of  given  length  and  diameter  in  feet  (Lifting 
weight  of  water  not  included). 

I  m  q8  m  q8 

~".616853i/dTT  =  .616853  X  ^d11  ^  *     ) 

To  find  the  pressure  lost  by  friction  while  discharging  a 
given  quantity  in  cubic  feet  per  second: — 


SULLIVAN'S  NEW  HYDRAULICS.  103 

n/ 


P'=.6mWdii (45) 

The  length  of  pipe  in  feet  through  which  a  given  pres- 
sure in  pounds  per  square  inch  will  force  a  given  quantity 
of  discharge  is, 

P/d"  X  .61685 

mq2 

To  find  the  coefficient  of  resistance  n,  in  terms  of  q,  d 
and  P:— 


. 61685 


To  find  the  coefficient  of  velocity   in  terms   of  q,  d   and 
^~ (48) 

**1 

18.— Formulas  In  Terms  of  Hydraulic  Radius   (r)   and 
Slope  (S). 

m  I  v8         m  I  v8 


Hy/r3       Hry/r 

mv*  mv* 


mv»  mv*    ' (5°) 

Hrv/r       Sv/r8  S 

m=-T7"~=-^~=  -v^Xv^3 (51) 


h"rvT         8 

I  v8     =  ~  vs      ~~    v 


=  ^1-Xl/r" I 


The  length  in  which  there  must  be  a  fall  of   one  foot  in 
order  to  generate  any  given  mean  velocity  v,  is 


^=^-=-m72- <») 


/TT      /     m^  \  /H       T/r3  /Sr/r« 

v=i/  H^  ("vr^  i/  —x  ^-=  v-^~- 

V**  X  \/-^r (54) 


104  SULLIVAN'S  NEW  HYDRAULICS. 

Area  in  square  feet  =12.5664Xr2.     q=av. 

q=12.5664r»Xe/'8Xl//-a7 (55) 

qT/m^f/riiXi/S  X  12.5664,  whence, 
i  X  157.91 44 


q2 


m=— 

nJLi^=.  ...(57) 

S"="l57.9U4Vrrf"=    1573144"  X  7^ (58) 

S=157.914Vri'i (59) 

H=i57:9WFn- «»> 


157.9144^11 

m 


*8 


....(61) 
...(62) 


-v'  - '  T7'91i4 <«) 


'-V   SBSOSg-xB*—  ••'(64> 


117          q*m2 

V   249\36:958"XS2"                             '"^ 
^157^/rii  ^ 

19.— Application  and  Limitations  of  the  Foregoing 
Formulas.—  As  heretofore  noted  under  the  table  of  circles 
and  of  open  channels  (§3)  r  is  not  necessarily  an  index  of  ca- 
pacity in  open  channels  as  it  is  in  pipes  and  circular  channels 
flowing  full.  Hence  in  open  channels  the  formulas  (63)  and 
(55)  for  q  will  not  necessarily  give  accurate  results,  unless  the 
value  of  r  was  originally  determined  in  terms  of  q  when  the 


SULLIVAN'S  NEW  HYDRAULICS.  105 

channel  was  designed.  In  channels  having  side  slopes  of  2  to 
1  the  formula  for  q  will  usually  apply  quite  accurately.  For 
the  same  reason  the  formula  for  r  does  not  apply  to  open 
channels  in  general,  but  only  to  those  in  which  the  value  of  r 
was  determined,  or  is  to  be  determined  in  terms  of  q.  All  the 
formulas  apply  with  exactness  to  pipes  and  circular  closed 
channels  flowing  full.  All  the  formulas  in  terms  of  r,  except 
those  just  noted,  apply  to  all  forms  of  open  or  closed  chan- 
cels. These  exceptions  in  the  case  of  open  channels,  do  not, 
however,  affect  the  general  application  of  the  coefficients,  be- 
cause the  coefficient  depends  upon  the  relations  of  a  to  p 
which  is  always  expressed  in  any  given  case  by  r  which  is 

equal  —  in  all  cases,  and  the  friction  surface  p,  always    bears 

P 

the  same  relation  to  r  that  r*  bears  to  the  area  in  any  given 
case  whether  r  is  an  index  of  capacity  or  of  length  of  peri- 
meter or  not. 

20— Formulas  in  Which  C  is  Used  Instead  of  m. 

In  the  tables  of  coefficients  heretofore  developed  the 
values  of  both  m  and  C  were  given  in  order  that  either  form 
of  the  formula  might  be  applied  in  any  case  at  pleasure.  All 
the  formulas  using  the  coefficient  C  instead  of  m  may  be  de- 
duced from  the  following: — 

v=C  t/r  !/r  /S (67) 

v=C  f/r  t/rS (68) 

v=C  e/i8  i/S=C1/S7':I" (69) 

v=C  t/d  !/d  v/S (70) 

v=C  e/d  !/dS". (71) 

v=C  t/d8  1/S=C1/S7dr, (72) 

v=C  S/d8  !/?-!// : (73) 

v=C  £/r3  /H  -r-  i/l (74) 

Area  in  square  feet,  A  =  d8  X  -354 

Area  in  square  feet,  A  =  r8  X  12.5664 

q=Av.    The  same  limitations  mentioned  in  the  preced- 


106  SULLIVAN'S  NEW  HYDRAULICS 

ing  section  will  also  be  observed  in  the  formulas  in  this  form 
relating  to  q  and  r  in  open  channels. 

FORMULAS  IN  TERMS  OF  DIAMETER  IN  FEET  USING  C. 
By  transposition  in  formula  (72)  we  have, 


v*-c  t/d8  " 

0 

"•(YD) 

J-CVd8 

r     /"HZ      v 

•••(?()) 

•~V    S!/d8  ~  ^3X^8 
M 

••  (70 

02Xi/d8 
C2H  v/d8 

•••(78) 

q  =  ds  X  -7854  XCX    t/d8  X    i/S  =  t/d11 
X-7854                (80) 

•  •  (79) 
v/S  X 

t/a»           q          di'         q4 

C^S-X.7854'             S2C*.3805 

,    11  /        q4 

21.— Formulas  in  Terms  of  Hydraulic  Radius  in  Feet 
Using  C. 

By  transposition  in  formulas  (69)  and  (74)  we  have 

•(82) 


C=l/S7F 


•  8= 


H= 


C«  H 


q=Av=12.5664Xr8XCX*/r8  X 

t/r^xcysxia.sees 


SULLIVAN'S  NEW  HYDRAULICS,  107 


11  /  q4 

I/  24936.958XC4 


"  157.9144 
I  q8 


7.9144  "  "<(90) 

C*1/r^    157.9144  " 
|=  157.9144     y"0'H  ........................  (92) 

C==  125664  ,/SX  t/r11 

A  set  of  formulas  in  terms  of  pressure  in  pounds  per 
square  inch  and  diameter  in  feet  or  inches  may  be  deduced 
in  like  manner  from  equation  (73).  It  is  not  deemed  neces- 
sary to  deduce  the  formula  in  all  its  possible  forms  and 
terms,  as  that  is  a  simple  matter  which  may  be  performed 
at  the  pleasure  of  the  person  using  it,  and  would  require  un- 
necessary space  here. 

22.—  Special  Formula  for  Vertical  Pipes.—  Because 
of  the  relation  of  H  to  I  in  all  formulas  the  ordinary 
formulas  for  flow  will  not  apply  to  a  pipe  in  a  vertical 
or  nearly  vertical  position.  In  such  case  H  and  I  in- 

TT 

crease    at    tne    same     rate,     and     hence  -y=l,     regardless 

of  the  head  or  length.  On  account  of  this  fact  all  the 
formulas  of  the  different  writers  on  hydraulics  will  give 
the  same  velocity  for  a  head  of  a  hundred  feet  as  for  a  head 
of  1,000  feet.  It  is  therefore  necessary  to  use  a  special  form- 
ula in  such  case.  In  a  vertical  pipe  the  water  is  supported 
at  no  point  whatever  by  any  portion  of  the  pipe  walls.  The 
effect  of  gravity  is  not  impeded  except  by  the  roughness  of 
the  pipe  walla.  In  such  vertical  pipe  there  is  a  gain  of  one 
foot  head  for  each  foot  of  length.  The  resistance  to  entry 
and  the  pipe  wall  friction  will  be  the  only  loss  of  head.  Hence 


108  SULLIVAN'S  NEW  HYDRAULICS. 

if  the  sum  of  their  effects  be  deducted  from  the  total  head, 
the  velocity  should  equal  that  due  to  the  remainder  of  the 
head.  On  this  theory  the  following  formula  is  proposed: 


TH-  (-=£- 


-) 


The  head,  slope,  velocity,  or  quantity    may  be    found   by 
the  principles  given  in  §  52,  and  table  No.  24. 


CHAPTER  IV. 


Of  Tables  for  Rapid  Calculation  of  Velocity  and  Discharge 
in  Open  and  Closed  Channels,  Friction  Loss,  &c. 


23.    Table  for  Velocity  and  Discharge.     Clean,  Average 
Weight  Cast  Iron  Pipes,  Not  Coated.— In  tables  No.   1  and 
No.  2  the  diameters  are  given  in  inches,  the  areas  in  square 
feet,  and  the  discharge  in  cubic  feet  per  second. 
How  to  use  Tables  No.  1  and  No.  2. 

To  find  the  mean  velocity  in  feet  per  second:— Multiply 
the  quantity  in  column  No.  5  opposite  the  given  diameter  in 
inches  by  ,/sT  For  v/S^  see  table  No.  15,  §  30.  For  S,  see 
Table  No.  16,  §  31. 

To  find  the  discharge  in  cubic  feet  per  second:— Multiply 
the  quantity  in  column  No.  6  opposite  the  given  diameter  by 
,/S. 

v=C  X  V&  v/~ST    q  =  AV=  AC  X  t/ds"l/sT     Take 
d  in  inches. 

For  average  weight  clean  cast  iron  pipe,  C  =  7.756  when 
d  =  inches. 


SULLIVAN'S  NEW  HYDRAULICS. 

TABLE.  No.  1. 
Clean  cast  iron  pipe,  not  coated. 


109 


Col.  1 
Diam. 
Inch's 

Col.  2 
Inches 

Col.  3 

t/d8 

Inches 

Col.  4 
Area 
Sq.  Feet 

Col.  5 
For 
Velocity 

Col.  6 
For 
Discharge 
ACXK  d8 

Vt 
& 

0.35355 
0.65227 

0.5946 
0.8059 

.001366 
.003068 

4.6117 
6.2505 

.006300 
.019176 

1.00 

1. 

1. 

.005454 

7.7560 

.042301 

1.34 

1.3975 

1.1820 

.008522 

9.1675 

.078125 

1.8360 

1.3550 

.01227 

10.5093 

.128949 

i'.£ 

2.3152 

1.5210 

.01670 

11.7968 

.197006 

2 

2.8284 

1.6810 

.02232 

13.0378 

.291003 

3 

5.1961 

2.2790 

.04909 

17.6759 

.867709 

I 

8. 

2.8284 

.08726 

21.9370 

1.91422 

5 

11.1803 

3.3439 

.13630 

25.9352 

3.53496 

6 

14.6969 

3.8340 

.19635 

29.7365 

5.83876 

7 

18.5202 

4.3040 

.26730 

8.92295 

8 

22.6274 

4.7570 

.34910 

36  '.8952 

12.88014 

9 

27. 

5.1960 

.44180 

40.3001 

17.80458 

10 

31.6227 

5.6231 

.54540 

43.6127 

23.78636 

11 

36.4828 

6.0400 

.66000 

46.8462 

30.91849 

12 

41.5692 

6.4470 

.7854 

50.0029 

39.27000 

13 

46.8721 

6.8460 

.9218 

53.0975 

48.94527 

14 

52.3832 

7.237 

1.069 

56.1301 

60.00307 

15 

58.0747 

7.622 

1.227 

59.1162 

72.53557 

16 

64. 

8. 

1.396 

62.0480 

86.61900 

17 

70.0927 

8.372 

1.576 

64.9332 

102.3347 

18 

76.3675 

8.738 

1.767 

67.7719 

119.7529 

19 

82.8190 

9.100 

1.969 

70.5796 

138.8300 

20 

89.4427 

9.457 

2.182 

73.3484 

160.0462 

21 

96.2340 

9.810 

2.405 

76.0863 

182.9876 

22 

103.189 

10.158 

2.640 

78.7854 

208.0000 

23 

110.304 

10.504 

2.885 

81.4690 

235.0380 

24 

117.575 

10.844 

3.1416 

84.1060 

264.2274 

25 

125. 

11.180 

3.409 

86.7120 

295.6012 

26 

132.574 

11.514 

3.687 

89.3025 

329.2583 

27 

140.296 

11.844 

3.976 

91.8620 

365.2433 

28 

148.162 

12.172 

4.276 

94.4060 

403.6800 

29 

156.169 

12.496 

4.587 

96.9189 

444.5670 

30 

164.316 

12.820 

4.909 

99.4319 

488.1112 

31 

172.600 

13.139 

5.241 

101.9061 

534.0898 

32 

181.0193 

13.456 

5.585 

104.3647 

582.8768 

33 

189.5705 

13.768 

5.940 

106.7846 

634.3005 

36 

216.0000 

14.698 

7.069 

113.9977 

805.8497 

40 

252.8822 

15.907 

8.726 

123.3747 

1076.5676 

44 

291.8629 

17.086 

10.558 

132.5190 

1399.1356 

48 

332.5537 

18.237 

12.567 

141.4461 

1777.5531 

54 

396.8173 

19.920 

15.905 

154.4995 

2457.3145 

60 

464.7580 

21.560 

19.635 

167.2193 

3283.3509 

72 

606.9402 

24.710 

29.607 

191.6507 

5674.2022 

84 

769.8727 

27.746 

38.484 

215.1979 

8281.6759 

96 

940.6040 

30.670 

50.265 

237.8765 

11956.8622 

120 

1314.5341 

36.250 

78.540 

281.1550 

22081.9137 

REMARK. — In  large  cast  iron  pipes,  or  in  thick  small  pipes 
there  is  great  liability  to  blow  holes  and   rough  places.    The 


110 


SULLIVAN'S  NEW  HYDRAULICS. 


thicker  the  pipe  shell  is,  the  more  liable  it  is  to  be  rough.  It 
might  be  well  to  take  C=7.65  in  terms  of  diameter  in  inches 
for  cast  iron  pipes  of  48  inches  diameter  or  greater,  and  for 
other  and  smaller  diameters  that  are  equally  thick  as  48  inch 
pipe.  Large  pipes  are  never  as  perfect  or  as  smooth  as  med- 
ium diameters  and  thicknesses. 

This  fact  has  led  some  engineers  to  conclude  that  the  law 
of  friction  in  pipes  was  slightly  different  in  large  pipes  from 
what  it  is  in  medium  diameters.  It  is  claimed  that  this 
change  occurs  at  about  a  diameter  of  48  inches.  It  is  due 
simply  to  the  rougher  casting  of  large  pipes  which  require 
thickness.  There  is  no  change  in  the  law  of  friction  at  any 
diameter  whatever.  Very  small  cast  iron  pipes  are  also  cast 
thick  to  prevent  breakage  in  handling  and  are  usually  as 
rough  as  the  very  large  diameters.  Pipes  less  than  six  inches 
diameter  and  over  36  inches  diameter,  are  usually  rougher  to 
some  extent  than  the  intermediate  diameters. 

q=A  ~~ 

TABLE  No.  2. 


Asphaltum    coated   pipes.    C=8.b7,    d=inches. 

Col.  1 

Col.  2 

Col.  3 

Col.  4 

Col.  6 

Col.  6 

Diam 

s~fta 

$/~d* 

Area 

cxt/in^ 

ACX£/7F*- 

Inch's 

Inches 

Inches 

Sq. 

For 

For 

Feet 

Velocity. 

Discharge 

6 

14.6969 

3.834 

.19635 

33.2407 

6.42681 

7 

18.5202 

4.304 

.2673 

37.3156 

9.97445 

8 

22.6274 

4.757 

.3491 

41.2432 

14.39800 

9 

27. 

5.196 

.4418 

44.9493 

19.85860 

10 

31.6227 

5.623 

.5454 

48.7522 

26.58945 

11 

36.4828 

6.040 

.6600 

52.3668 

34.56208 

12 

41.  5692 

6.447 

.7854 

55.8955 

43.90032 

13 

46.8721 

6.846 

.9213 

59.3548 

54.71325 

14 

52.3832 

7.237 

1.069 

62.7448 

67.07419 

15 

58.0747 

7.622 

1.227 

66.0827 

81.08347 

16 

64. 

8. 

1.396 

69.3600 

96.82656 

17 

70.0927 

8.372 

1.576 

72.5852 

lit.  39427 

18 

76.3675 

8.738 

1.767 

75.7584 

133  .  86509 

19 

82.8190 

9.100 

1.969 

78.8970 

155.34819 

20 

89.4427 

9.457 

2.182 

81.9922 

178.90698 

21 

96.2340 

9.810 

2.405 

85.  '527 

204.55174 

22 

103.189 

10.158 

2.640 

88.0698 

232.50443 

23 

110.304 

10.504 

2.885 

91.0697 

262.73603 

24 

117.575 

10.844 

3.1416 

94.0L74 

295.36541 

25 

125. 

11.180 

3.409 

96.9306 

330.43641 

26 

132.574 

11.514 

3.687 

99.8264 

368.05986 

27 

140.296 

11.844 

3.976 

102.6874 

407.88542 

28 

148.162 

12.172 

4.276 

105.4312 

450.82398 

29 

156.169 

12.496 

4.587 

108.3403 

496.957C4 

30 

164.316 

12.820 

4.909 

111.1494 

545.63240 

SULLIVAN'S  NEW  HYDRAULICS. 


Ill 


TABLE  No.  2— Continued. 


Col.  1 

Diam. 
Inch's 

Col.  2 
v/d8 
Inches 

Col.  3 

e/ds 

Inches 

Col.  4 
Area  Sq. 
Feet. 

Col.  5 
CXt/d8 
Velocity 

Col.  6 

ACXf/d8 
For  Disch'g. 

31 

172.600 

13.139 

5.241 

113.9151 

597.02919 

32 

181.0193 

13.456 

5.585 

116.6635 

651.56576 

33 

189.5705 

13.768 

5.940 

119.4685 

709.64324 

34 

198.2523 

14.081 

6.305 

122.0822 

769.72871 

35    ' 

207.U62S 

14.390 

6.681 

124.7613 

833.52624 

86 

216. 

14.698 

7.069 

127.431B 

900.80267 

87 

225.0622 

15.002 

7.467 

130.0673 

963.93283 

38 

234.2477 

15.300 

7.876 

132.6510 

1044.75927 

40 

252.8822 

15.907 

0    f-OC 

8.  /26 

137.9137 

1203.43495 

44 

291.8629 

17.086 

10.558 

148.1356 

1564.01587 

48 

332.5537 

18.237 

12.567 

158.1148 

1987.02869 

54 

3%.  8173 

19.920 

15.905 

172.7064 

2746.89529 

M 

464.7580 

21.560 

19.635 

186.9252 

3670.27630 

72 

6f«!94<r> 

24.710 

29.607 

214.2357 

6342.87637 

84 

769.8727 

27.746 

38.484 

240.5478 

9257.24164 

96 

940.6040 

80.670 

50.265 

265.9089 

1S365.  41086 

REMARK— This  table  relates  to  asphaltum  coated  pipes— 
not  to  pipes  coated  with  coal  tar,  nor  to  compound  coatings 
made  of  only  one  part  asphaltum.  What  is  meant  by  as- 
phaltum coated  pipes  is  that  class  of  pipes  which  have  been 
properly  coated  with  a  compound  composed  of  18  to  20  per 
cent  of  crude  petroleum  and  the  remainder  of  asphaltum. 
The  coating  compound  to  be  heated  to  300  degrees,  Fahr., 
and  the  pipe  to  remain  submerged  in  the  hot  bath  until  the 
pipe  metal  attains  the  same  temperature  as  that  of  the  bath. 
Coal  tar  coatings  do  not  form  quite  as  smooth  a  surface  as 
the  above  described  coating,  and  hence  do  not  develop  as 
high  values  of  C.  If  d  is  taken  in  feet,  then  m =.00032,  and 
C=55.90  as  the  average  value  of  the  coefficients  for  asphal- 
tum  and  oil  coated  pipes.  The  value  of  C  or  m  will  vary 
slightly  with  the  quality  or  purity  of  the  asphaltum  used. 
(See  group  No.  2.) 

24.— Table  for  Velocity  and  Discharge  of  Brick  Lined 
Circular  Conduits  or  Sewers  Flowing  Full.— In  the  follow- 
ing Table  No.  3  the  diameters  are  in  feet,  the  areas  in  square 
feet,  and  the  discharge  in  cubic  feet  per  second.  The  coeffi- 
cient is  in  terms  of  diameter  in  feet  and  is  based  upon  the 
discharge  of  Washington,  D.  C.,  aqueduct,.  (See  Group  6.) 


112 


SULLIVAN'S  NEW  HYDRAULICS. 


;or    v= 


m=.0008577;     C= 


34.00  in  terms  of  diameter  in  feet.     q=A  CXt/d*  >/! 

TABLE  No.  3 
Circular  brick  conduits  and   sewers.     C=34.00. 


Col.  1 
Diana. 
Feet 

Col.  2 
v/d3 
Feet 

Col.  3 

t/ds 
Feet 

Col.  4 
Area 
Sq  Feet 

Col.  5 
For  Vel. 
CXt/d" 

Col.  6 
For  Disch'g 
ACXt/ds 

1.50 

1.837 

1.355 

1.767 

46.070 

81.4057 

2.00 

2.8-28 

1.681 

3.142 

57.154 

179.5778 

2.50 

3.953 

1.988 

4.909 

67.592 

331.8091 

3.00 

5.196 

2.279 

7.068 

77.486 

537.6711 

4.00 

8. 

2.828 

12.566 

96.166 

1208.8016 

5.00 

11.180 

3.344 

19.635 

113  6% 

2232.4210 

6.00 

14.697 

3>34 

28.274 

130.356 

3685.6855 

7.00 

18.5^0 

4.304 

38.485 

146.336 

5631.7410 

8.00 

22.627 

4.757 

50  266 

161.738 

8129.9223 

9.00 

27. 

5.1% 

63.617 

176.664 

11238.8337 

10.00 

31.623 

5.623 

78.540 

191.182 

15015.4343 

11.00 

36.4b3 

6.040 

95.033 

205.360 

19515.9769 

12.00 

41.569 

6.447 

113.100 

219.198 

24791.2938 

13.00 

46.872 

6.846 

132.730 

232  764 

30894  .  7657 

14.00 

52.38< 

7.237 

153.940 

246.058 

37878.1685 

NOTE.— Compare  the  values  of  the  coefficients  of  the  new 
Croton  aqueduct  for  a  depth  of  9  feet  with  those  of  the  Wash- 
ington aqueduct— both  in  group  No.  6.  The  above  value  of 
C  in  terms  of  diameter  in  feet  is  about  correct  for  plain  brick. 

25.— Egg  Shaped  Brick  Sewers  and  Conduits.— IK  egg 

shaped  sewers  the  vertical  diameter  is  one  and  one-half  times 
the  horizontal  or  greatest  transverse  diameter.  Radius  of  in- 
vert, %  vertical  diameter.  Radius  of  sides  equal  vertical  diam- 
eter. Let  d=greatest  transverse  diameter  in  feet. 

a=area  in  square  feet. 

p=wetted  perimeter  in  lineal  feet. 

r=hydraulic  mean  depth=L 
P 
Then,  in  egg  shaped  sewers  and  conduits, 

a=d*X-284  for  \  full  depth 

a=d8X-755825  for  f  full  depth. 

a=d*Xl-148525  for  full  depth. 
The  wet  perimeter  in  lineal  feet  will  be, 

p=dXl-3747  for  \  full  depth. 


SULLIVAN'S  NEW  HYDRAULS.  113 

p=dX23941  for  |  full  depth. 
p=dX3.965  for  full  depth. 

The  mean  hydraulic  depths, '  r~=r,  will  be, 

r=dX-2066  for  £  full  depth. 

r=dX-3157  for  |  full  depth. 

r=dX-2897  for  full  depth. 

See  "Hydraulic  Tables"  by  P.  J.  Plynn;  Van  Nostrand'e  Sci- 
ence Series  No.  67,  and  also  see  "Treatise  on  Hydraulics"  by 
Prof.  Merriman,  p.  235.  (5th.  Edition.) 

TABLE     FOR     VELOCITIES      AND       DISCHARGES    OP    EGG    SHAPED 

BRICK    CONDUITS    AND   SEWERS    PLOWING  TWO-THIRDS 

FULL    DEPTH. 

As  this  class  of  conduits  are  not  circular  in  form,  the 
coefficient  is  in  terms  of  hydraulic  mean  depth  (r)  in  feet, 
and  the  value  of  the  coefficient  used  in  the  following  table 
is  that  developed  by  the  Washington,  D.  C.,  aqueduct,  (See 
Group  No.  6).  This  table  is  to  be  used  in  the  same  manner 
as  Tables  Nos.  1,2  and  3. 

v=CX  t/r8  ,/S,  and  q=ACXt/r8  i/S.    C=96.00 
TABLE  No.  4. 

Areas,  hydraulic  depths,  velocities  and  discharges  for  % 
Full  Depth.  0=96. 


Col.  0. 

Col.  ICol.  1 

Col.  2 

Col.  4 

Column  5. 

Column   C 

Trans. 
Diam. 

r 
Feet 

j/r' 

Feet 

J£ 

Area 
Sq.Ft 

For  velocity 
CXKr8 

For  Dischg 
AC  X  V** 

1.50 

0.474 

0.3263 

0.5713 

1.701 

54.8443 

93.2910 

2.00 

0.631 

0.5'U2 

0.70.O 

3.025 

67.9680 

205.6032 

2.50 

0.789 

0.7008 

0.8372 

4.724 

80.3712 

379.3535 

3.00 

0.947 

0.9216 

0.0600 

6.802 

92.1600 

626.8723 

3.50 

1.105 

1.1615 

1.0780 

9.259 

103.4880 

958.1954 

4.00 

1.263 

1.419 

1.19LO 

12.093 

114.3360 

1382.6653 

4.50 

1.421 

1.694 

1.302 

15.305 

124.9920 

1913.0025 

5.00 

1.579 

J.984 

1.408 

18.895 

135.1680 

2555  .  3994 

6.00 

1.894 

2.606 

1.614 

27.210 

154.9440 

4216.0262 

7.00 

2.210 

3.285 

1.812 

37.035 

173.9520 

6442.3123 

8.00 

2.526 

4.015 

2.004 

48.373 

192.3840 

9306.1912 

9.00 

2.841 

4.789 

2.188 

61.222 

210.0480 

12859.5586 

10.00 

3.157 

5.610 

2.368 

75.583 

227.3280 

17182.1322 

11.00 

3.473 

6.472 

2.544 

91.455 

244.2240 

22335.5059 

Small  sewers  should  be  circular  in  form.    See  §55. 


114  SULLIVAN'S  NEW  HYDRAULICS. 

26—  Formulas  for  Use  in    Connection  With   the  Fore- 
going Tables. 

In  the  tables  for  pipes  and  conduits  are   the  tabular  val- 
ues of 

CXKd8,  CX^r8,  A  C  XS/d*  and  A 
Now 


If  the  slope  and  mean  velocity  have    been   decided  upon, 

y 

then   the    value  of  CX^/d8=—  TQ-  and  opposite  this  value  of 


V* 

is  the  required  diameter  to  generate  the  given  veloc- 
ity. 

If  a  given  diameter  is  required  to  discharge  a  given 
number  of  cubic  feet  per  second,  then  the  grade  or  slope  may 
be  found  thus: 

/«  q 

V  b—  A  CXf/d» 

The  grade  to  generate  a  given  velocity  in  feet  per  second 
may  be  found  thus: 


If  the  quantity  to  be  discharged  and  the  grade  are  given, 
then  the  required  diameter  will  be  found  thus: 

ACXv/d^yg.  Look  for  the   diameter  which  corresponds 

to  the  value  of  A  CXv/'d8  in  the  table. 

The  general  formulas  already  given  are  so  simple  that  re- 
sort to  these  formulas  is  not  necessary. 


SULLIVAN'S  NEW  HYDRAULICS.  115 

27— General  Table  of  Values  of  r  or  d,  With  Roots- 

TABLE  No.  5 


r  or  d 

V^r  or^/d 

y'r8  or  v/d» 

^'r»  or  t/d§ 

0.20 

0.4472 

0.089440 

0.2990 

.22 

.4690 

.103180 

.3212 

.24 

.4899 

.117576 

.3429 

.26 

.5099 

.132574 

.3641 

1 

.5291 

.5477 

.148148 
.164310 

.3849 
.4053 

.32 

.5656 

.180992 

.4244 

.34 

.5831 

.19S254 

.4452 

.36 

.6000 

.216000 

.4647 

.38 

.6164 

.234232 

.4840 

.40 

.6324 

.252960 

.5030 

.42 

.6481 

.272202 

.5217 

.44 

.6633 

.291852 

.5402 

.46 

.6782 

.311972 

.5585 

.48 

.6928 

.332544 

.5767 

.50 

.7071 

•354550 

.5946 

.52 

.7211 

.374972 

.6124 

.54 

.7348 

.396792 

.6299 

.56 

.74<<3 

.419048 

.6473 

.58 

.7616 

.441728 

.6646 

.60 

.7746 

.464760 

.6817 

.62 

.7874 

.488188 

.6987 

.64 

.8 

.512 

.7155 

.66 

.8124 

.536184 

.7322 

.68 

.8246 

:  560728 

.7488 

.70 

.8366 

!  585620 

.7653 

.72 

.8485 

.610920 

.7816 

.74 

.8602 

.636548 

.7978 

.76 

.8718 

.662568 

.8139 

.78 

8832 

.688896 

8300 

.80 

!8944 

.715520 

.8459 

.82 

.9055 

.742510 

.8617 

.84 

.9155 

.769020 

.8774 

.86 

.9273 

.797478 

.8930 

.88 

.9380 

.825440 

.9086 

.90 

.9487 

.853830 

.9240 

.92 

.9591 

.882872 

.9394 

.94 

.96 

.9695 
.9798 

.911330 

.940308 

.9546 
.9698 

.98 
1.00 

.9899 
1. 

.970102 
1. 

.9849 
1. 

1.02 

1.010 

1.0302 

1  015 

1.04 

1.020 

1.0600 

1.029 

1.06 

1.029 

1.0907 

1.045 

1.08 

1.039 

1.1521 

1  059 

.10 

1.049 

1.1540 

1.074 

.12 

1.058 

1.1850 

1.089 

.14 

1.068 

1.2175 

1.104 

.16 

1.077 

1.2  94 

1.118 

.18 

1.086 

1.2815 

1.132 

.20 

1.095 

1.3140 

1.146 

.22 

1.104 

1.3469 

1.160 

.24 

1.114 

1.3800 

1.175 

.26 

1.123 

1.4150 

1.189 

116  SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  No.  5. — Continued. 


r  or  d 

y/r  or  i/d 

v/r8  or  !/d« 

e/r"  or  f/d« 

1.28 

131 

1.4480 

.203 

1.30 

.144 

1.4872 

.217 

1.32 

.150 

1.5180 

.239 

1.35 

.161 

1.5673 

.252 

1.40 

.183 

1.6562 

.287 

1.45 

.214 

1.7603 

.326 

1.50 

.225 

1.8375 

.355 

1.55 

.245 

1.9297 

.399 

1.60 

.265 

2.0240 

.422 

1.65 

.284 

2.1186 

.455 

1.70 

.304 

2.2168 

.488 

1.75 

.323 

2.2852 

.511 

1.80 

.341 

2.4138 

.554 

1.85 

.360 

2.5160 

.568 

1.90 

.378 

2.6182 

.618 

1.95 

.396 

2.7122 

.647 

2. 

.414 

2.8284 

.663 

2.05 

.431 

2.9335 

.713 

2.10 

.459 

3.0639 

.750 

2.15 

.466 

3.1519 

.775 

2.20 

.483 

3.2626 

.806 

2.25 

.500 

3.3750 

.837 

2.30 

.526 

3.5098 

.873 

2.35 

.533 

3.6025 

.904 

2.40 

.549 

3.7176 

.928 

2.45 

.565 

3.8342 

1.958 

2.50 

.581 

3.9525 

1.988 

2.55 

.597 

4.0723 

2.018 

2.60 

.612 

.1912 

2.047 

2.65 

.638 

.3407 

2.083 

2.70 

.643 

.4361 

2  106 

2.75 

.658 

.5595 

2.135 

2.80 

.673 

.6844 

2.164 

i:S 

.688 
.703 

.8108 
.9387 

2.193 
2.222 

2.95 

.717 

5.0651 

2.250 

3. 

.732 

5.1960 

2279 

3.05 

.746 

5.3253 

2.308 

3.10 

.761 

5.4591 

2.336 

3.15 

.775 

5.5912 

2  364 

3.20 

:  .789 

5.7248 

2.392 

3.25 

5.8597 

2.421 

3.30 

!816 

5.9928 

2.448 

3.35 

.830 

6.1305 

2.476 

3.40 

.844 

6.2696 

2.504 

3.45 

.857 

6.4066 

2.531 

3.50 

.871 

6.5485 

2.559 

3.55 

.884 

6.6882 

2.586 

3.60 

.897 

6.8292 

2.613 

3.65 

.910 

6.9715 

2.640 

3.70 

.923 

7.1151 

2.667 

3.75 

.936 

7.26  0 

2.694 

3.80 

.949 

7.4062 

2.721 

3.85 

.962 

7  5537 

2.746 

3.90 

.975 

7.7025 

2.775 

3.95 

1.987 

7.8486 

2.801 

SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  NO.  5.— Continued. 


117 


r  or  d 

y'r  or  y'd. 

,/f  8  or  ,/d8 

t/r8  or  e/d8 

4. 

2. 

8. 

2.828 

4.05 

2.012 

8.1486 

2.854 

4.10 

2.024 

8.2984 

2.881 

4.15 

2.037 

8.4535 

2.908 

4.20 

2.049 

8.6058 

2.933 

4.25 

2.061 

8.7592 

2.960 

4.30 

2.073 

8.9139 

2.986 

4.35 

2.085 

9.0698 

3.012 

4.40 

2.097 

9.2268 

3.037 

4.45 

2.109 

9.3850 

3.064 

4.50 

2.121 

9.5445 

3.089 

4.55 

2.133 

9.7051 

3.115 

4.60 

2.144 

9.8624 

3.140 

4.65 

2.156 

10.0254 

3.165 

4.70 

2.168 

10.1896 

3.192 

4.75 

2.179 

10.3502 

3.218 

4.80 

2.191 

10.5168 

3.243 

4.16 

2.202 

10.6797 

3  268 

4.90 

2.213 

10.8437 

3.293 

4.95 

2.225 

11.0137 

3.319 

5. 

2.236 

11.1800 

3.344 

5.05 

2.247 

11.3473 

3.369 

5.10 

2.258 

11.5158 

3.393 

5.15 

2.269 

11.6853 

3.419 

5.20 

2.280 

11.8560 

3.444 

5.25 

2.291 

12.0277 

3  468 

5.30 

2.302 

12.2006 

3  493 

5.35 

2.313 

12.3745 

3.518 

5.40 

2.823 

12.5442 

3.542 

5.45 

2.334 

12.7203 

3  567 

5.50 

2.845 

12.8975 

3.592 

5.55 

2  356 

13  0758 

3  616 

5.60 

2.366 

13.2496 

3^640 

5.65 

2.377 

13.4300 

3  665 

5.70 

2.388 

13.6116 

3.689 

5.75 

2.398 

13.7885 

3.713 

5.80 

13.9664 

3  737 

5.85 

2  412 

14.1453 

3.761 

5.90 

2^429 

14.3311 

3.786 

5.95 

2.439 

14.5120 

3  810 

6. 

2.449 

14.6940 

3.834 

6.05 

2.460 

14.8830 

3  858 

6.10 

2.470 

15.0670 

3  881 

6.15 
6.20 

2.480 
2.490 

15.2520 
15.4380 

31905 
3  929 

6.25 

2.500 

15.6250 

3  953 

6.30 
6.35 

2.510 
2.520 

15.8130 
16.0020 

3.977 
4.000 

6.40 

2.530 

16.1920 

4  024 

6.45 

2.540 

16.3830 

4  047 

6.50 

2.550 

16.5750 

4.071 

6.55 

2.560 

16.7680 

4  094 

6.60 

2.569 

16.9554 

4l  118 

6.65 

2.579 

17  .  1503 

4  140 

6.70 

2.588 

4  164 

6.75 

2.598 

1715365 

4.188 

118  SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  No.    5— Continued. 


r  or  d 

l/r  or  \/d 

j/r*  or  -j/d8 

V*  or  yd* 

6.80 

2.607 

17.7276 

4.211 

6.85 

2.617 

17.9264 

4.234 

6.90 

2.627 

18.1263 

4.257 

6.95 

2.636 

18.3202 

4.280 

7. 

2.645 

18.5150 

4.304 

7.05 

2.655 

18.7177 

4.327 

7.10 

2.665 

18.9215 

7.15 

2.6H 

19.1191 

4^373 

7.20 

2.683 

19.3176 

4.395 

7.25 

2.692 

19.5170 

4.418 

7.30 

2.702 

19.7246 

4441 

7.35 

2.711 

19.9258 

4.464 

7.40 

2.720 

20.1280 

4.487 

7.45 

2.729 

20.3310 

4.510 

7.50 

2.739 

20.5425 

4.532 

7.55 

2.748 

20.7474 

4.555 

7.60 

2.756 

20.9456 

4.578 

7.65 

2.766 

21.1600 

4.600 

7.70 

2.775 

21.3675 

4.622 

7.75 

2.784 

21.5760 

4.642 

7.80 

2.793 

21.7854 

4.668 

7.85 

2.802 

22.0000 

4.690 

7.90 

2.811 

22.2069 

4.712 

7.95 

2.819 

22.411Q 

4.735 

8. 

2.828 

22.6240 

4.759 

8.05 

2.837 

22.8378 

4.779 

8.10 

2.846 

23.0526 

4.801 

8.15 

2.855 

23.2682 

4.823 

8.20 

2.864 

23.4848 

4.846 

8.25 

2.872 

23.6940 

4.868 

8.30 

2.881 

23.9123 

4.890 

8.35 

2.890 

24.1315 

4.912 

8.40 

2.898 

24.3432 

4.934 

8.45 

2.907 

24.5641 

4.950 

8.50 

2.915 

24.7775 

4.978 

8.55 

2.924 

25. 

5 

8.60 

2.932 

25.2152 

5.013 

8.65 

2.941 

25.4396 

5.044 

8.70 

2.949 

25.6563 

5  066 

8.75 

2.958 

25.8825 

.088 

8.80 

2.966 

26.1008 

.109 

8.85 

2.975 

26.3300 

.131 

8.90 

2.983 

26.5487 

.153 

8.95 

2.992 

26.7784 

.175 

9. 

3. 

27. 

196 

9.05 

3.008 

27.2224 

'217 

9.10 

3.016 

27.4456 

5  239 

9.15 

3.025 

27.6787 

5  261 

9.20 

3.033 

27.9036 

5^283 

9.25 

3.041 

28.1293 

5.304 

9.30 

3.049 

28.3*57 

9.35 

3.058 

28.5923 

5i348 

9.40 

3.066 

28.8204 

5  369 

9.45 

3.074 

29.0493 

5.390 

9.50 

3.082 

29.2790 

5.411 

9.55 

3.091 

29.5190 

5.433 

SULLIVAN'S  NEW  HYDRAULICS. 
TABLE  No.  5— Continued. 


119 


r  or  d 

y'r  or  y'd 

y'r8  or  -j/d8 

f/r«  or  f/d8 

9.60 
9.65 
9.70 
9.75 
9.80 
9.85 
9.90 
9.95 
10. 

3.098 
3  106 
3.114 
3.123 
3.131 
3.138 
3.146 
3.154 
3.162 

29.7408 
29.9729 
30.2058 
30.4492 
30.6838 
30.9093 
81.1454 

31  !  6200 

5.454 
5.475 
5.496 
5.518 

5^560 
5.581 
5.602 
5.623 

28.— Tables  for  Velocity  and  Discharge  of  Trapezoidal 
Canals.  In  Pig.  1,  let  A,  E,  F,  D,  equal  the  width  of  the 
water  surface  in  feet.  Let  B  C  equal  bottom  width  of  canal 
in  feet,  and  E  B  or  P  C,  equal  greatest  depth  of  water  in 
feet. 

TO    FIND   THE   AREA    IN   SQUARE   FEET. 

Multiply  E  D  by  E  B,  or  F  A  by  B1  C.  Or  secondly:  Add 
together  the  width  of  water  surface  and  the  bottom  width  in 
feet,  and  divide  the  sum  by  2.  Then  multiply  the  quotient 
by  the  depth  F  C  or  E  B  in  feet.  In  either  case  the  result 
will  equal  the  area  in  square  feet. 

TO    FIND   THE   LENGTH  A  E   OR   F  D   IN   FEET. 

If  the  side  elopes  A  B  and  D  C  are  1  to  1,  then  AE=E  B» 


120  SULLIVAN'S  NEW  HYDRAULICS. 

and  P  D=F  C.  If  the  side  slopes  are  \yz  horizontal  to  1  vert, 
ical,  then  A  E=E  BXl-50.  If  the  side  slopes  are  2  horizontal 
to  1  vertical,  then  A  E=E  BX2.00. 

TO   FIND   THE   WETTED   PERIMETER   IN   LINEAL   FEET. 

The  length  of  B  C,  or  of  the  bottom  width  in  feet,  is,  of 
course,  always  known.  It  is,  therefore,  only  required  to  find 
the  length  in  feet  of  the  side  slopes  A  B  and  D  C  which  when 
added  to  B  C,  will  equal  the  total  wetted  girth  or  perimeter- 
If  the  side  slopes  are  1  to  1,  then  the  length  A  B  or  D  C  is 
equal  to  the  diagonal  of  a  square,  or  equal  to  the  depth  of 
water  E  BX  1.41421. 

The  length  of  either  side  slope  for  any  rate  of  slope  what- 
ever is  the  same  as  the  hypotenuse  of  a  right  angled  triangle, 

and  A  B=V(AE)8+(E  B)8  or  D  C=v/(F  D)8+(P  C)8. 

Adding  together  the  lengths  in  A  B,  B  C,  and  C  D,  we 
have  the  wetted  perimeter  (p)  in  feet.  The  hydraulic  mean 

depth  in  feet  is  then  r=  area  ip  Bquare  feet  =4 
wet  perimeter  in  feet     p 

In  the  following  tables  of  trapezoidal  canals  the  value  of 
the  area  in  square  feet,  and  the  hydraulic  mean  depth  r,  and 
of  $/r3  for  each  additional  half  foot  depth  of  water  in  each 
canal  is  given,  so  that  the  velocity  and  discharge  for  each 
depth  of  flow  may  be  readily  ascertained.  The  value  of  m  or 
C  will  depend  upon  the  material  forming  the  wetted  perimeter, 
and  the  condition  of  the  canal  a&  to  good  or  bad  repair.  The 
value  of  m  or  G  may  be  selected  from  the  tables  of  values  de- 
veloped in  the  groups  of  rivers  and  canals  heretofore  given. 
The  following  tables  show  the  area  for  each  depth  of  water. 
The  discharge  for  any  given  depth  will  equal  the  area  for  that 
depth  multiplied  by  the  mean  velocity.  The  slope  required  to 
generate  any  desired  mean  velocity  in  feet  per  second  for  any 
depth  of  flow  will  be 


SULLIVAN'S  NEW  HYDRAULICS. 


121 


The  distance  or  length  in  feet  (I)  of  canal  in  which  there 
must  be  a  fall  of  1  foot  in  order  to  generate  a  given  mean  vel- 
ocity in  feet  per  second  may  be  found  by  the  formula, 


HOW  TO  USE  THE  FOLLOWING  TABLES. 

To  find  the  mean  velocity  for  any  given  depth,   multiply 

t/r8  for  that  depth  by  <V S,  or  multiply  t/r8  by  J— 

Vm 

To  find  the  discharge  in   cubic   feet  per  second  for  any 
given  depth  of  flow,  multiply  t/r3  by  areaXCVS,  or  multiply 

t/r8  by  areaX-J-.    For  values  of  v/S,  see  §  30,  Table  No.  15. 

\  m 


tol. 


Trapezoidal  canal. 


TABLE  No.  6. 
Bottom  width  2  feet. 


Side  slopes  1 


Depth  of 
Water  in  ft. 

Area 
Square  Feet 

Feet 

V/r« 
Feet 

t/r« 
Feet 

1.00 
1.50 
2.00 
2.50 
3.00 

3.00 
5.25 
8.00 
11.25 
15.00 

0.6213 
0.8400 
1.0450 
1.240 
1.430 

0.4897 
0.7699 
1.068 
1.380 
1.710 

0.6998 
0.8774 
1.034 
1.177 
1.308 

REMARK.— In  climates  where  the  earth  freezes  in  winter, 
side  slopes  of  earth  will  not  stand  if  they  are  steeper  than  1% 
to  1  even  in  very  firm  earth.  In  lighter  soil  in  frosty  climates 
the  side  slopes  should  vary  from  1%  to  1  to  3  to  1,  according  to 
the  nature  of  the  soil. 


122  SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  No.  7. 
Trapezoidal  canal.    Bottom  width  4  feet.    Side  slopes  1  to  1. 


Depth  of 
Water  in  ft. 

Area 
Square  Feet 

Feet 

V*3 
Feet 

t/r« 
Feet 

l.°0 

5.00 

0.732 

0.6263 

0.7914 

1.50 

8.25 

.000 

1.000 

.000 

2.00 

12.00 

.2426 

1.385 

.176 

2.50 

16.25 

.4700 

1.782 

.335 

3.00 

21.00 

.762 

2.339 

.530 

.50 

26.25 

.888 

2.594 

.611 

.00 

32.00 

2.089 

3.019 

.737 

.50 

38.25 

2.280 

3.443 

1.855 

.00 

45.00 

2.480 

3.905 

1.976 

.50 

52.25 

2.670 

4.383 

2.089 

6.00 

60.  00 

2.861 

4.836 

2.199 

6.50 

68.25 

3.050 

5.327 

2.308 

7.00 

77.00 

3.235 

5.819 

2.412 

8.00 

96.00 

3.605 

6.844 

2.616 

TABLE  No.  8. 
Trapezoidal  canal.  Bottom  width  4  feet.    Side  slopes  1*^  to  1. 


Depth  of 
Water  in  ft 

Area  Square 
Feet 

Feet 

3/r« 

Feet. 

f/r" 
Feet. 

1.00 

5.500 

0.7231 

0.6149 

0.7841 

1.50 

9.375 

.0000 

1.0000 

.000 

2.00 

14.000 

.2488 

1.3960 

.181 

2.50 

19.375 

.4880 

1.815 

.347 

3.00 

25.500 

.7200 

2.256 

.502 

3.50 

32.375 

.8370 

2.490 

.578 

4.00 

40.000 

2.1710 

3.099 

.788 

4.50 

48.375 

2.4000 

3.718 

.928 

5.00 

57.500 

2.6100 

4.216 

2.053 

6.00 

78.000 

3.0420 

5.306 

2.303 

7.00 

101.500 

3.4730 

6.472 

2.544 

8.00 

128.000 

3.9000 

7.702 

2.775 

Trapezoidal  canal. 


TABLE  No.  9. 
Bottom   width  6  feet. 


Side  slopes  1  to  1. 


Depth  of 
Water  in  ft. 

Area  Square 
Feet 

Feet 

Felt. 

Feet. 

1.00 

7.00 

0.7929 

0.7060 

0.8403 

1.50 

11.25 

1.0980 

1.1510 

.0725 

2.00 

16.00 

1.3726 

1.6090 

.2680 

2.50 

21.25 

1.6280 

2.077 

.4420 

3.00 

27.00 

1.8639 

2.545 

.595 

3.50 

33.25 

2.0900 

3.022 

.738 

4.00 

40.00 

2.3100 

3.511 

.874 

4.50 

47.25 

2.5200 

4.000 

2.000 

5.00 

52.00 

2.5800 

4.144 

2.036 

5.50 

63.25 

2.9340 

5.025 

2.242 

6.00 

72.00 

3.1345 

5.548 

2.355 

7.00 

91.00 

3.5270 

6.624 

2.574 

8.00 

112.00 

3.9100 

7.731 

2.780 

SULLIVAN'S  NEW  HYDRAULICS. 


123 


Trapezoidal  canal. 


TABLE  No.  10. 
Bottom  width  8  feet. 


Side  elopes  2  to  1. 


Depth  of 
Water   Feet 

Area 
Square  Feet 

Feet 

V/r' 
Feet 

Feet 

1.00 

10.00 

0.8018 

0.7179 

0.8473 

1  50 

16.50 

1.1210 

1.187 

1.089 

2.00 

24.00 

1.4164 

1.685 

1.298 

2.50 

32.50 

1.7000 

2.216 

1  489 

3  00 

42.00 

1.9600 

2.744 

1.656 

3  50 

52.50 

2.2200 

1.819 

4.00 

64.00 

2.470 

3  882 

1.970 

4.50 

76.50 

2.720 

4^486 

2.118 

5.00 

90.00 

2.964 

5.103 

2.259 

6.00 

120.00 

3.445 

6.394 

2.528 

7.00 

154.00 

3.910 

7.732 

2.781 

Trapezoidal  canal. 


TABLE  No.  11. 
Bottom  width  8  feet. 


Side  slopes  1  to  1. 


Depth  of 
Water    Feet 

Area 
Square  Feet 

r 
Feet 

£ 

fr/r« 
Feet 

1.00 

9.00 

0.831 

0.7576 

0.8704 

1.50 

14.25 

1.164 

1.256 

1.121 

2.00 

20.00 

1.464 

1.771 

1.331 

2.50 

26.25 

1.741 

2.297 

1.516 

3.00 

33.00 

2.000 

2.828 

1.682 

3.50 

40.25 

2.248 

3.370 

4.00 

48.00 

2.485 

3.917 

1.979 

4.50 

56.25 

2.710 

4.461 

2  112 

5.00 

65.00 

2.935 

5.029 

2^242 

5.50 

74.25 

3.153 

5.598 

2.366 

6.00 

84.00 

3.364 

6.170 

2.484 

7.00 

105.00 

3.777 

7.341 

2.709 

Trapezoidal  canal. 


TABLE  No.  12. 
Bottom  Width  10  feet. 


Side  elopes  1  to  1 


Depth  of 
Water  Feet 

Area 
Square  Feet 

Fe'et 

Feet 

Feet 

1.00 

11.00 

O.a574 

0.7939 

0.891 

2.00 

24.00 

1.5320 

1.896 

1.377 

2.50 

31.25 

1.8300 

2.475 

1.574 

3.00 

39.00 

2.1100 

3.065 

1.750 

3.50 

47.25 

2.3743 

3.658 

1.912 

4.00 

56.00 

2.6270 

4.258 

2.063 

4.50 

65.25 

2.8700 

4.862 

2.205 

5.00 
5.50 

75.00 
85.25 

3.1060 
3.3350 

5.474 
6.090 

2^468 

6.00 

96.00 

3.5600 

6.717 

2.591 

7.00 

119.00 

3.9930 

7.979 

2.825 

8.00 

144.00 

4.4130 

9.271 

3.045 

124 


SULLIVAN'S  NEW  HYDRAULICS. 


Trapezoidal  canal. 
2tol. 


TABLE  No.  13. 
Bottom   width  10  feet. 


Side  elopes 


Depth  of 
Water  Feet 

Area 
Square  Feet 

Fe^t 

Feet 

t/r" 
Feet 

1.00 

12.00 

0.8222 

0.7455 

0.8634 

2.00 

28.00 

1.478 

1.797 

1.340 

2.50 

37.50 

1.774 

2.363 

1.537 

3.00 

48.00 

2.049 

2.933 

1.713 

3.50 

59.50 

2.320 

3.534 

1.880 

4.00 

72.00 

2.581 

4.146 

2.036 

4.50 

&5.50 

2.840 

4.786 

2.187 

5.00 

100.00 

-3.093 

5.440 

2.332 

5.50 

115.50 

3.340 

6.104 

2.471 

6-00 

132.00 

3.584 

6.785 

2.605 

7.00 

168.00 

4.068 

8.493 

2.914 

8.00 

208.00 

4.543 

9.683 

3.111 

29.— Table  for  Velocity  and  Discharge  of  Rectangular 
Channels,  Flumes,  Masonry  Conduits  etc.—  The  value  of  the 
coefficient  to  be  used  with  the  following  table  will  depend 
upon  the  nature  and  condition  of  the  lining  of  the  flume  or 
channel.  According  to  the  experiments  of  D'Arcy  and  Bazin, 
the  average  value  of  C  for  unplaned  board  flumes,  well  joint- 
ed, and  without  strips  or  battens  on  the  inside  is  C=U9.00,  or 
m=.00007.  For  nicely  dressed  lumber  flumes,  well  jointed 
and  without  battens  on  the  inside,  their  experiments  give  C 
=128.00  as  an  average.  If  we  refer  to  the  last  two  flumes  in 
Group  No.  5,  one  at  Boston  gives  C=106.30,  and  the  High- 
line  in  Colorado  gives  C=70.00.  The  data  of  flow  in  wooden 
conduits  are  very  unsatisfactory.  The  density  of  the  wood, 
the  closeness  of  joints,  the  alignment  of  the  flume,  gritty  de- 
posits etc..  all  affect  the  value  of  C  in  any  case.  Where  the 
flume  is  constructed  of  rough,  very  knotty,  lumber  and  has 
battens  on  the  inside  to  cover  the  joints,  it  is  proable  that 


SULLIVAN'S  NEW  HYDRAULICS. 


125 


the  value  of  the  coefficient  will  be  about  C=80.00,  if  the 
alignment  of  the  flume  is  fairly  direct.  For  channels  lined 
with  brick,  ashlar,  rubble  etc.,  see  the  groups  of  such  chan- 
nels for  value  of  C.  See  table  No.  15  for  value  of  /S. 


TABKE  No.  14 
Flumes  and  other  rectangular  channels. 


Width 
Feet 

Depth    of 
Water    feet. 

Area 

Sq.  Feet. 

r 

Feet. 

Feet. 

t/r3 
Feet. 

1.5 

0.50 

0.75 

0.300 

0.1643 

0.4054 

•    1.5 

1.00 

1.50 

.429 

0.2810 

0.5301 

2.0 

0.75 

1.50 

.429 

0.2810 

0.5301 

2.0 

1.50 

3.00 

.600 

0.4647 

0.6817 

3.0 

1.00 

3.00 

.600 

0.4647 

0.6817 

3.0 

1.50 

4.50 

.750 

0.6495 

0.8059 

3.0 

2.00 

6.00 

.860 

0.7975 

0.8930 

4.0 

1.50 

6.00 

.860 

0.7975 

0.8930 

4.0 

2.00 

8.00 

1.000 

1.0000 

1.0000 

5.0 

1.50 

7.50 

.937 

0.9070 

0.9524 

5.0 

2.00 

10.00 

.111 

.171 

1.082 

5.0 

3.00 

15.00 

.363 

591 

.261 

6.0 

2.00 

12.00 

.200 

.314 

.147 

6.0 

2.50 

15.00 

.363 

.591 

.261 

6.0 

3  00 

18.00 

.500 

.837 

.355 

6.0 

4.00 

24.00 

.714 

2.244 

.498 

8.0 

3.00 

24.00 

.714 

2.244 

.498 

8.0 

4.00 

32.00 

2.000 

2.828 

.682 

8.0 

5.00 

40.00 

2.222 

3.312 

.820 

8.0 

6.00 

48.00 

2.400 

3.718 

1.928 

10.0 

4.00 

40.00 

2.222 

3.312 

1.820 

10.0 

5.00 

50.00 

2.500 

3.953 

1.988 

10.0 

6.00 

60.00 

2.727 

4.503 

2.122 

10.0 

7.00 

70.00 

2.916 

4.979 

2.231 

12.0 

4.00 

48.00 

2.400 

3.718 

1.928 

12.0 

5.00 

60.00 

2.727 

4.503 

2.122 

12.0 

6.00 

72.00 

3.000 

5.196 

2.279 

12.0 

7.00 

84.00 

3.230 

5.805 

2.410 

14.0 

5.00 

70.00 

2.916 

4.979 

2.231 

14.0 

6.00 

84.00 

3.230 

5.805 

2.410 

30.     Table  of  Values  of  Slopes  S  and  ^S. 

The  distance  in  feet  I,  in  which  there  is  a  fall  of  one  foot  is 


126  SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  No.  15. 
Value  of  S  and  i/S. 


Fall  per 
Mile  in  Feet 

Fall 
One  In 

Slope 
S 

•s 

0.50 

10560.00 

.0000947 

.009731 

0.75 

7042.25 

.0001420 

.011915 

1.00 

5280.00 

.0001894 

.013762 

1.76 

3000.00 

.0003333 

.018255 

2.00 

2640.00 

.0003788 

.019463 

2.64 

2000.00 

.0005000 

.022361 

3.00 

1760.00 

.0005682 

.023836 

3.18 

1660.00 

.0006024 

.024544 

3.30 

1600.00 

.0006250 

.025000 

3.38 

1560.00 

.0006410 

.025318 

3.52 

1500.00 

.0006667 

.025820 

3.62 

1460.00 

.0006849 

.026171 

3.70 

1427.00 

.0007007 

.026472 

3.75 

1408.00 

.0007102 

.026650 

3.80 

1389.00 

.0007199 

.026832 

8.85 

1371.00 

.0007294 

.027007 

3.90 

1354.00 

.0007385 

.027176 

4.00 

1320.00 

.0007576 

.027524 

.20 

1257.00 

.0007955 

.028205 

.40 

1200.00 

.0008333 

.028868 

.50 

1173.00 

.0008525 

.029198 

.60 

1148.00 

.0008710 

.029514 

.70 

1123.00 

.0008905 

.029841 

.75 

1111.00 

.0009000 

.030001 

.80 

1100.  00 

.0009090 

.030151 

4.90 

1078.00 

.0009276 

.030457 

5.00 

1056.00 

.0009469 

.030773 

5.10 

1035.00 

.0009662 

.031083 

5.20 

1015.00 

.0009852 

.031388 

5.28 

1000.00 

.0010000 

.031623 

6.00 

880.00 

.0011364 

.033710 

7.00 

754.30 

.0013258 

.036411 

8.00 

660.00 

.0015151 

.038925 

9.00 

586.60 

.0017044 

041286 

10  00 

528.00 

.0018940 

.043519 

11.00 

480.00 

.0020833 

.045643 

12.00 

440.00 

.0022730 

.047673 

13  00 

406.10 

.0024621 

.049620 

14.00 

377.10 

.0026515 

.051493 

15.00 

352.00 

.0028409 

.053300 

16  00 

330.00 

.0030303 

.055048 

17.00 

310.60 

.0032197 

.056742 

18.00 

293.30 

.0034090 

058388 

19.00 

277  TO 

.0035985 

.059988 

20  00 

264.00 

.0037878 

.061546 

21.00 

251.40 

.0039773 

.063066 

22.00 

240.00 

.0041666 

.064549 

23.00 

229.60 

.0043560 

.066000 

24.00 

220.00 

.0045454 

.067419 

25.00 

211.20 

.0047348 

.068810 

26. 

203.10 

.0049242 

.070173 

27. 

195.20 

.0051136 

.071510 

SULLIVAN'S  NEW  HYDRAULICS. 
TABLE  No.  15.— Continued. 


127 


Fall  Per 
Mile  in  Feet 


Fall, 
One  In 


Slcgpe 


51. 

52. 
52.80 
55.80 
60.00 
70.00 
80.00 
90.00 
100.00 
120.00 
140.00 
160.00 
180.00 
200.00 
240.00 
280.00 
320.00 
360.00 
400.00 
450.00 
500.00 


700.00 
800.00 


188.60 
182.10 
176.00 
170.30 
165.00 


155.30 
150.90 
146.60 
142  70 


135.40 

132.00 

128.80 

125.70 

122.80 

120.00 

117.30 

114.80 

112.30 

110.00 

107.70 

105.60 

103.50 

101.50 

100.00 

96.00 

88.00 

75.43 

6600 

58.66 

52.80 

44.00 

37.71 

33.00 

?933 

26.40 

22.00 

18.86 

16.50 

14.66 

13.20 

11.73 

10.56 

880 

7.54 


.0053030 

.U  ).->-!  '.KM 
.0056818 
.0058712 


.0062500 
.0064394 


.0068182 

.0070075 


.0075757 
.0077651 
.0079545 


.0085227 
.0087121 


.0098485 

.01 

.0104167 


.0132576 
.0151515 
.0170455 


.022727 
.0265151 
.0303030 


.0378787 
.0416667 


.0681818 
.0757575 
.0852273 


.1136364 
.1325757 
.1515151 


.072822 
.074111 
.075378 


.081417 
.082572 
.083711 


.085944 
.087039 
.088120 
.089188 
.090244 


.097312 

.098281 

.099241 

.10 

.102060 

.106600 

.115141 


.137620 
.150756 


.174077 
.184637 
.194625 


.261116 

.275241 


.307729 
.337100 
.364109 


31 — Table  of  Slopes  tor  Average  Weight  Clean  Cast 
Iron  Pipes,  Showing  the  Inclination  Required  in  Each  Di- 
ameter to  Generate  a  Mean  Velocity  of  One  Foot  per  Sec- 


128  SULLIVAN'S  NEW  HYDRAULICS. 

and,  from  which  the  Slope  Required  to  Generate  any 
other  Mean  Velocity  may  be  Found. 

f—Hdt/d    jf  m  tl3js  generai  formula  we  assign  H— 1  foot 
mv* 

and  v*=l  foot,  we  have,  2=1^1  as   the  formula  for   finding 

m 

the  length  in  which  there  must  be  a  total  head,  fall  or  slope  of 
one  foot  to  generate  a  velocity  of  one  foot  per  second.  For 
this  class  of  pipe  m  is  a  constant,  and  in  terms  of  diameter  in 
feet  m=.0004,  or  in  terms  of  diameter  in  inches  m=.0004X 
V/(l?)3=.01662768.  Hence  the  length  in  feet  I,  in  which  there 
must  be  a  head  or  fall  of  one  foot  in  order  to  generate  a  mean 
velocity  of  one  foot  per  second  will  be 

I  =1/dS  if  d  is  taken  in  feet,  and  s= 
.0004 


The  length  in  feet  in  which  there  must  be  a  head  or  fall 
of  one  foot  in  order  to  generate  any  given  or  desired  mean 
velocity  in  feet  per  second  is, 

d-^/d      i/d8  m  v*  /m  v* 

In  which  vs  is  the  square  of  the  given  or  desired  velocity 
in  feet  per  second.  The  coefficient  m  may  be  in  feet  or  in 
inches  as  above  but  the  mean  velocity  will  be  in  feet  per 
second  in  either  case. 

The  required  slope  S,  to  generate  a  mean  velocity  of  one 
foot  is, 

m  m 

S=  •  ,  j3  ,  and  to  generate  any  velocity  is  S=    ,  ~3  X  v*. 

Hence  if  the  slope  for  any  diameter,  which  causes  v=l 
be  taken  from  the  following  table,  the  required  slope  to 
cause  any  other  velocity  may  be  found  at  once  by  multiply- 
ing this  slope  from  the  table  by  the  square  of  the  desired 
velocity,  v*. 


SULLIVAN'S  NEW  HYDRAULS.  12§ 

EXAMPLE. 

From  Table  No.  16  it  is  seen  that  a  slope  of  S=.0004  for 
a  pipe  12  inches  diameter,  will  generate  a  mean  velocity  of 
one  foot  per  second.  Required,  the  slope  of  a  12  inch  pipe  to 
generate  5  feet  per  second  velocity: 

SOLUTION;— From  Table  16,  take  the  slope  for  1  foot 
velocity,  S— .0004.  Multiply  this  slope  by  the  square  of  the 
required  velocity,  and  we  have, 

S=.0004X(o)8=.01,and  l=-^-=  ~=  100  feet.    In  other 

words  there  must  be  a  fall   of  one  foot   in   a   length   of   100 
feet. 

TABLE  No.  16. 

Table  giving  the  required  slope  to  generate  a  mean 
velocity  of  one  foot  per  second  in  average  weight  clean  cast 
iron  pipes. 

.016628  1         T/d« 

For  v=l,        S= 


Diameter 
Inches 

/d8 
Inches 

S 

Diameter 
Inches. 

T/d* 
Inches 

S 

3 

5.1961 

.003200000 

26 

132.5740 

.0001254242 

4 

8. 

.002078500 

27 

140.2960 

.0001185208 

5 

11.1803 

.001487250 

28 

148.1620 

.oooir^x, 

6 

14.6969 

.001131156 

29 

156.1690 

.0001064743 

7 

18.5202 

.0008975830 

30 

164.3160 

.  000101  •-i.XX) 

8 

22.6274 

.000734861 

31 

172.6000 

.OCO(i<v-:j,:>:;:> 

9 

27. 

.0006158&2 

32 

181.0193 

10 

31.6227 

.000525825 

33 

189.5705 

!OOOU8771.«>I} 

11 

36.4828 

.0004=57764 

34 

198.2523 

.000083S0790 

12 

41.5692 

.000400000 

35 

207.0628 

.OOUO!S<-:»'4]4 

13 

46.8721 

.000354752 

36 

216. 

.00007f.'.ixl(K) 

14 

52.3832 

37 

225.0822 

.000073SXI70 

15 

58.0747 

^  000286320 

234.2477 

.00007098(80 

16 

64. 

.000260000 

40 

252.8222 

.00006.-75390 

17 

70.0927 

.000237228 

44 

291.8629 

18 

76.3675 

.0002177366 

48 

332.5537 

JQBKOOOOM 

19 

82.8190 

.0002007753 

54 

396.8173 

.  00004  1'.»':!41 

20 

89.4427 

.0001859069 

60 

464.7580 

.0000357777*5 

21 

96.2340 

.0001727870 

72 

606.9402 

.0000274.  «IO 

22 

103.1890 

.0001611411 

84 

769.8727 

.00002160000 

23 

110.3040 

.0001516536 

96 

940.6040 

:000017r,7,xm 

24 

117.5750 

.0001414254 

120 

1314.5341 

.00001264935 

to 

125. 

.0001330240 

130  SULLIVAN'S  NEW  HYDRAULICS. 

32.—  Head  in  Feet  Lost  by  Friction  in  Average  Weight 
Clean  Cast  Iron  Pipes  for  Different  Velocities  of  Flow. 

By  equation  (10)  the  coefficient  of  resistance  or  friction  is 
b"  di/d     S" 


(10) 

From  which  the  formula  for  head   lost  by   friction  h",  is 
n  I  v8       n  I  v*       n 

h"=-d7o-=-7d^=v^x'v'  .................  <lb) 

For  a  constant  diameter  and  velocity  the  friction  loss 
will  be  directly  as  the  length  in  feet  (  I  )of  pipe,  and  will  vary 
as  v*  for  different  velocities.  For  constant  degrees  of 
roughness  of  pipe  n  is  a  constant 

As  the  friction  loss  is  inversely  as  |/d8  and  directly  as  the 
length  and  as  v*,  the  loss  in  one  foot  length  of  any  diameter 

when  v2=l,  will  be  S"  =    /  ^8  and  for  any  other    velocity    it 

u 
will  be  S"=  /  jg  Xv>aQd  for  any    length  in   feet    of  pipe  it 

ii 
will   be  /j8X*Xv*.  Hence  if  we  form  a  table  which  shows  the 

loss  of  head  in  feet  for  one  foot  length  of  pipe  and  for  a  velocity 
of  one  foot  per  second,  the  loss  for  any  other  length  in  feet 
will  be  found  by  multiplying  the  tabular  quantity  by  the 
given  length  in  feet  /,  and  the  loss  for  any  velocity  will  be 
found  by  multiplying  by  the  square  of  that  velocity,  v*.  (See 
§  9  and  §  10.) 

TABLE  No.  17. 

Table  showing  the  loss  of  head  in  feet  by  friction  in  one 
foot  length  of  clean  cast  iron  pipe  when  v*=l. 


SULLIVAN'S  NEW  HYDRAULICS.  131 

.01637 

-,    when 


When  vz=l,  the  loss  per  foot  length  =-i= 
d=incheg. 


Diam. 

Inchet 

T/d3 
Inches. 

Bead  lost  in 
!eet  per  foot 
length 

Diam. 
Inches 

;/d8 
Inches. 

Hd  lost  in  ft 
per  ft  length 

3 

5.1961 

.003150440 

25 

125. 

.00013U96000 

4 

8. 

.002046250 

26 

132.5740 

.00  '12347820 

5 

11.1803 

.0014641820 

27 

140.2960 

.000116681710 

6 

14.6969 

.0011138327 

28 

148.1620 

.000110487170 

7 

18.5202 

.0008839000 

29 

156.1690 

.000104822340 

8 

22.6274 

.0007234590 

30 

164.3160 

.OOU0996251126 

9 

27. 

.0006062963 

31 

172.6000 

.(XX  '094843569 

10 

31.6227 

.0005176661 

32 

181.0193 

.1  00(190435330 

11 

36.4828 

.000448704CO 

33 

189.5705 

.000'  186353094 

12 

41.5692 

.0003937900 

34 

198.2523 

.000082571551 

13 

46.8721 

.00034924806 

35 

207.0628 

.000079058190 

14 

52.3832 

.000312:.(!177L' 

36 

216. 

.000075787037 

15 

58.0747 

.000281878341) 

37 

225.0622 

.000072735444 

16 

64. 

.  000255  7M  •-':>(  i 

38 

234.2477 

.000069883290 

17 

70.0927 

.  0002335  47.V.M 

40 

252.8822 

.0-10064  730000 

18 

76.3675 

.000214358200 

•44 

291.8629 

.00005608800 

19 

82.8190 

.00019766  OCO 

48 

332.5537 

.0000492  '3100 

20 

89.4427 

.000183022203 

54 

396.8173 

.000041253239 

21 

96.2340 

.00017016620(1 

60 

464.7580 

.000035222632 

22 

103.1890 

.00015Si;iU!'4(l 

72 

606.9402 

.0000270000  0 

23 

110.3(MO 

.00014810NW 

84 

769.8727 

.000  212632556 

117.5750 

.000139230278 

96 

940.6D40 

.  (10001  74037000 

REMARK. — As  the  loss  here  tabulated  is  for  one  foot 
length  only  and  for  a  velocity  of  only  one  foot  per  second,  none 
of  the  decimals  should  be  cut  off  especially  in  case  the  pipe  is 
of  considerable  length  and  the  velocity  is  high,  because  the 
losa  increases  directly  as  the  number  of  feet  in  length  and 
also  asv8. 

How  TO  USE  TABLE  No.  17. 

The  table  shows  the  loss  of  head  in  feet  by  friction  for 
oue  foot  length  of  pipe  of  each  diameter,  and  for  a  velocity  of 
one  foot  per  second.  If  the  pipe  is  several  hundred  feet  in 
length,  then  move  the  decimal  point  two  places  to  the  right- 
This  will  be  equivalent  to  multiplying  by  100,  and  will  show 
the  loss  of  head  in  feet  per  100  feet  length  for  v"=l.  Mul- 
tiply this  result  by  the  square  of  the  actual  or  proposed  ve- 
locity in  feet  per  second  and  the  result  is  the  actual  loss  per 
100  feet  length  for  that  velocity.  If  the  pipe  is  several  thousand 


132  SULLIVAN'S  NEW  HYDRAULICS. 

feet  in  length  then  take  out  from  the  table  the  loss  for  one 
foot  length  and  v2  =1,  and  move  the  decimal  point  three 
places  to  the  right.  Multiply  by  the  square  of  the  actual  or 
proposed  velocity  in  feet  per  second.  The  result  will  be  the 
actual  loss  of  head  in  feet  per  1,000  feet  length  of  pipe.  The 
loss  of  head  in  feet  per  mile  (5280  feet)  of  pipe  equals  the  loss 
for  1,000  feet  multiplied  by  5.28. 

EXAMPLE. 

What  is  the  loss  of  head  in  feet  in  an  8  inch  cast  iron 
pipe  750  feet  in  length  when  the  velocity  is  six  feet  per  sec- 
ond ? 

SOLUTION. 

In  table  17,  opposite  a  diameter  of  8  inches  and  in  the 
third  column  the  tabular  loss  for  one  foot  length  of  8  inch 
pipe  when  vs=l  is  .000723459.  Multiplying  this  by  100  feet 
length  by  moving  the  decimal  point  two  places  to  the  right, 
and  the  loss  for  100  feet  =.0723459  when  vs=l.  As  the  act- 
ual velocity  is  to  be  six  feet  per  second,  and  as  the  loss  varies 
as  v2  in  any  given  diameter  the  last  result  must  be  multiplied 
by  (Q)s  =36,  and  we  have  the  actual  loss  per  100  feet  length= 
.0723459X36=2.60445  feet,  and  for  750  feet  the  loss  will  be 
2  60445X?  5=19  5334  feet. 

33.  —  Formula  and  Table  for  Ascertaining  the  Loss  of 
Head  in  Feet  In  any  Class  of  Pipe  While  Discharging  a 
Given  Quantity  In  Cubic  Feet  Per  Second. 

Let  h"=  total  head  in  feet  lost  by  friction  in  the  length 
I 

d=diameter  of  pipe  in  feet. 

q=  cubic  feet  per  second  discharged. 

Then  the  coefficient  of  resistance  is 

hyd"  X  .616853  _  Syd"X.616853 


Zq*  q* 

And  the  head  in  feet  lost  by  friction  is 


T-  <See  e<luation  32-> 


SULLIVAN'S  NEW  HYDRAULICS. 


133 


If  I  be  taken  =1  foot  length  of  pipe,  then   n   is  constant 
for  any  given  class  of  pipe,  and  we  may  take  the   quotient  of 

n  Q  ' 

•  gifigK    as  a  constant,  which,  when  multiplied  by— yfp  will 

equal  the  loss  of  head  in  feet  per  foot  of  pipe  for  the  given 
discharge  q.  As  qs  ond  Vs  are  convertible  terms  we  use  the 
same  coefficient  value  in  terms  of  either  q  or  v. 

The  value  of  n  in  terms  of  diameter  in  feet  for  ordinary 
cast  iron  pipes  is  n=.00039380. 

The  loss  of  head  in  one  foot  length  is   h"=   fi1ftQgQ      X 


Then- 


.00039380 


!/d"  '        '"  .616853 
h"  =  .OC063840  X: 


.616853 


=;.00063840.  Whence 


The  following    table   gives 


value  of^/d11 . 

The  slope  required  to  cause  a  given  diameter  to  discharge 

q  cubic  feet,  S=   ^353  X^p 

From    tables    Nos.    1  and  2,  q=a  cXt/d'Xv/S,  and  S= 


TABLE  No.  18. 
"  when  d  is  taken  in  feet.     (See  §  44.  45.) 


Values  of 


Diameter 
Inches 

Diameter 
Feet 

i/d" 

Feet 

Diameter 
Inches 

Diameter 
Feet 

v/d" 
Feet 

3 

0.2500 

.0004883 

24 

2.00 

45.25 

4 

0.3333 

.002375 

25 

2.083 

56.60 

5 

0.4167 

.00811 

26 

2.166 

70.17 

6 

0.5 

.0221 

27 

2.25 

86.50 

7 

0.5833 

.05157 

28 

2.333 

105.55 

8 

0.6667 

.1075 

29 

2.416 

128.00 

9 

0.75 

.2055 

30 

2.50 

154.40 

10 

0.8333 

.3668 

31 

2.584 

185.20 

11 

0.9167 

.6198 

32 

2.666 

219.90 

12 

1.0000 

1.000 

33 

2.75 

260.80 

13 

1.083 

1.55 

34 

2.834 

307.80 

14 

1.167 

2.338 

35 

2.916 

360.00 

15 

1.25 

3.412 

36 

3.00 

420.90 

16 

1.383 

4.859 

38 

3.166 

566.00 

17 

1.417 

6.800 

40 

3.333 

750.90 

18 

1.5 

9.301 

42 

3.50 

982.60 

19 

1.583 

12.51 

44 

3.666 

1268.00 

20 

1.667 

16.62 

48 

4.00 

2048.00 

21 

1.75 

21.71 

54 

4.50 

3914.00 

22 

1.833 

•28.01 

60 

5.00 

6979.00 

23 

1.917 

35.84 

72 

6.00 

19050.00 

134  SULLIVAN'S  NEW  HYDRAULICS. 

For  asphaltum  coated  pipes  take  n=. 000325  in   terms  of 
diameter  in  feet.    Then  for  such  coated  pipes, 

h"=-6^5lTX7Sl-X  l  =--00051864  X  ^TX  /  = 
.00051861 


q=cubic  feet  discharged  per  second. 
Z=length  of  pipe  in  feet. 
d=diameter  of  pipe  in  feet.     See  §  44. 

34.— Asphaltum  Coated  Pipes.  Table  for  Ascertaining 
the  Loss  of  Head  in  Feet  for  any  Velocity. 

By  formula  (16) 

n  I  v*        n  Z  n 

h  =  -avdr=7d-*=x  <V*=7H*XV 

The  average  value  of  n  for  this  class  of  pipe  is  n  =  . 00032 
in  terms  of  diameter  in  feet,  or  n=.013302  in  terms  of  diame- 
ter in  inches.  In  order  to  find  the  loss  of  head  in  feet  by 
friction  per  100  feet  length  of  pipe  for  any  velocity,  make  Z= 
100,  and  insert  the  value  of  n,  and  we  have 

1 00  X. 01 3302 
Head  lost  per  100  feet  length 


Xv^-V  rF~Xvs,  if  d  is  in  inches,  or'-::7^XvF,  if  d  is  in  feet. 
yd3  yd3 

TABLE  No.  19. 

Table  showing  loss  of  head  in  feet  per  100  feet  length  of 
asphaltum  coated  pipe  when  vs=l.    To  find  the  loss  for  any 


SULLIVAN'S  NEW  HYDRAULICS, 


135 


other  velocity  multiply  the  tabular  loss  by  the  square  of  that 
velocity  in  feet  per  second. 


Diam- 
eter 
In. 

T/d« 

Inches 

Head   in  Ft. 
Lost  per  100 
Feet 

Diam- 
eter 
In. 

!/d8 
Inches 

Head  in  Feet 
Lost  per  100 
Feet 

3 

5.1961 

.2560000 

23 

110.3040 

.0120594 

4 

8. 

.1662800 

24 

117.5750 

.01131363 

5 

11.1803 

.1190000 

25 

125. 

.01064160 

6 

14.6969 

.0905080 

26 

132.5740 

.01003364 

7 

18.5202 

.0718242 

27 

140.2960 

.009481382 

8 

22.6274 

.0588000 

28 

148.1620 

.009000000 

9 

27. 

.0492667 

29 

156.1690 

.008517700 

10 

31.6227 

.0420600 

30 

164.3160 

.008095377 

11 

36.4828 

.0364610 

31 

172.6000 

.0077068366 

12 

41.5692 

.0320000 

32 

181.0193 

.0073483880 

13 

46.8721 

.0283800 

33 

189.5705 

.0070700000 

14 

52.3832 

.0254000 

34 

198.2523 

.0067096300 

15 

58.0747 

.0229050 

35 

207.0628 

.0064241400 

16 

64.0000 

.02078436 

36 

216. 

.006160000 

17 

70.0927 

.01897770 

38 

234.2477 

.005680000 

18 

76.3675 

.01741840 

40 

252.8822 

.005460000 

19 

82.8190 

.01606140 

42 

272.2500 

.004885950 

20 

89.4427 

.01487200 

44 

291.8629 

.004557600 

21 

96.2340 

.01382250 

48 

332.5537 

.004000000 

22 

103.1890 

.01289000 

54 

396.8173 

.003352000 

What  is  the  loss  of  head  in  feet  by  friction  in  a  22  inch 
coated  pipe  2500  feet  in  length,  when  the  velocity  is  six  feet 
per  second? 

SOLUTION. 

From  table  19  we  see  that  the  loss  in  one  hundred  feet 
length  of  22  inch  pipe  is  .01289  feet  head  when  v«=l.  If  v=6, 
then  v*=36,and  .01289X36=.46404  feet  lost  per  100  feet  length 
of  pipe.  As  there  are  2,500  feet  of  pipe  the  total  loss  in  the 
whole  length  will  equal  the  loss  for  100  feet  length  multiplied 
by  the  number  of  100  feet,  or  25,  and  we  have  .46404X25= 
11.601  feet  head  lost  in  2500  feet  length  when  the  velocity  is 
6  feet  per  second. 

If  this  asphaltum  coated  pipe  were  replaced  by  an  aver- 
age weight  clean  cast  iron  pipe  22  inches  in  diameter,  what 
would  be  the  loss  of  head  in  the  cast  iron  pipe  for  6  feet  ve- 
locity, and  what  slope  would  be  required  to  cause  the  latter 
pipe  to  generate  6  feet  per  second  velocity? 
SOLUTION. 

From  table  No.  17  the  loss  of  head  per  one  foot  length  of 


136  SULLIVAN'S  NEW  HYDRAULICS. 

22  inch  cast  iron  pipe  when  va=l  is  .00015864094.  The  loss 
per  100  feet  =.015864094,  and  when  v=6  the  loss  per  100  feet 
will  be.015864094X(6)2=.571107384,  and  for  2500  feet, 
.571107384X25=14.2777.  The  slope  or  fall  in  the  2500  feet  must 
therefore  be  14.2777—11.601=2.6767  feet  greater  for  the  cast 
iron  pipe  than  for  the  aephaltum  coated  pipe. 

The  slope  in  either  pipe  which  is  required  to  generate  the 
given  velocity  is 


m=.0004  for  cast  iron 
m=.00033  for  asphaltum  coating 

These  values  of  m  are  in  terms  of  diameter  in  feet.  The 
value  of  m  may  be  converted  to  terms  of  diameter  in  inches 
by  multiplying  by  /(1  2)  8  =41.5692.  (See  §§  10,  12  and  Group 
No.  2,  §14.) 

The  slopes  to  generate  any  given  velocity  may  be  found 
from  Tables  No.  1  and  No.  2  by  the  formula 

s=(      -  -  s= 


Table  No.  18  gives  the  different  values  of  v/d11.  TaDles 
No.  1  and  2  give  the  values  of  f/d3  and  also  of  ACX  W  'or 
each  diameter  and  class  of  pipe.  When  d=feet,  the  slope  re- 
quired to  cause  a  cast  iron  pipe  to  discharge  a  given  number 
of  cubic  feet  per  second  q,  is 

S=  .000648456X-^r-     (See  §§  42,  43). 

From  which  the  diameter  in  feet  required  to  discharge  a 
given  quantity  when  the  slope  is  given,  is 

d=  |//.0000004205X^//^1,  for  clean  cast  iron,  or 


d=-i  /         1  when  d=inches. 

See  Tables  Nos.  1  and  2  for  value  of  AC,  and  see  formulas 


SULLIVAN'S  NEW  HYDRAULICS.  137 

35.— Plow  and  Friction  In  Fire  Hose.— Fire  hose  is  made 
of  different  material,  such  as  woven  hose,  lined  with  rubber, 
or  hose  made  entirely  of  leather.  The  resistance  to  flow  will 
depend  upon  the  nature  of  the  material  which  forms  the  lin- 
ing. The  resistance  to  flow  in  rubber  lined  hose  is  much 
smaller  than  in  leather  hose,  or  in  iron  pipes  of  equal  diam- 
eter. Fire  hose  of  all  classes  are  made  2^  inches  in  diame- 
ter, and  therefore  the  area  and  friction  surface  are  constan-t. 
Head  in  feet  and  pressure  in  Ibs  per  square  inch  increase  or 
vary  at  the  same  rate.  The  quantity  discharged  per  second 
by  a  hose  of  constant  diameter  increases  directly  as  the  ve- 
locity. In  a  constant  diameter  the  velocity  or  quantity  in- 
creases as  the  square  root  of  the  head  in  feet,  or  as  the 
square  root  of  the  pressure  in  Ibs  per  square  inch.  The 
friction  increases  as  va  or  q8  in  a  constant  diameter.  The 
pressure  or  the  head  is  as  va  or  qs.  The  coefficient  may  there- 
fore be  determined  in  terms  of  head  in  feet  or  in  terms  of 
pressure  in  Ibs  per  square  inch  and  in  terms  of  v*  or  qa.  The 
friction  loss  will  then  vary  as  the  head  or  pressure  or  as  vfl  or 
q1  in  the  constant  diameter.  As  fire  hose  are  all  2%  inches 
diameter,we  may  use  the  direct  value  of  the  coefficients  m  and 
n  instead  of  the  unit  values.  It  is  more  convenient  to  have 
the  discharge  of  fire  hose  in  gallons  per  minute  than  in  cubic 
feet  per  second,  hence  the  formulas  will  be  given  in  terms  of 
pressure  in  Ibs  per  square  inch  and  discharge  in  gallons  per 
minute. 

Let  P=total  guage  pressure  at  hydrant  or  steamer. 

P'^=total  pressure  lost  by  friction  in  the  length  I,  in  feet 
of  hose. 

q=gallons  per  minute  discharged  by  the   hose. 

n=coefficient  of  friction. 

As  the  diameter  is  constant,  the  direct  value  of  n  will  be 
P'  d  n 

n=rq^~'  and  p'  =-d~Xq5X  i. 

If  200  feet  of  rubber  lined  woven  hose  2^  inches  diame- 
ter be  laid  out  straight  on  a  level  with  one  end  attached  to  a 
hydrant  or  steamer,  and  with  a  smooth  nozzle  one  inch  chain- 


138  SULLIVAN'S  NEW  HYDRAULICS. 

eter  and  18  inches  in  length  at  the  other  end,  and  a 
pressure  guage  at  the  hydrant  end  registers  50  pounds, 
pressure  per  square  inch,  another  guage  attached  at  the  butt 
of  the  nozzle  on  the  other  end  will  register  only  35  Ibs  per 
square  inch,  and  the  discharge  will  be  145  gallons  per  minute. 
The  pressure  lost  in  the  200  feet  of  hose,  (not  including  the 
nozzle),  was  therefore  P'  =50—  35=15  Ibs.  Then, 


And  P'^'  Xq2  X  I  =.000003567Xq2  X  I 

q=  gallons  per  minute. 

Z=length  in  feet  of  hose. 

The  loes  of  pressure  in  Ibs  per  square  inch  in  2^  inch 
rubber  lined  woven  hose  of  any  length  and  for  any  discharge 
in  gallons  per  minute  will  therefore  be 

P'=.000003567Xq8XZ- 

In  experiments  with  this  class  of  hose  the  writer  has  ob- 
served that  the  friction  increases  very  slightly  for  low  pres- 
sures and  decreases  slightly  for  high  pressures,  be- 
cause as  the  pressure  within  the  hose  becomes  in- 
tense, the  rubber  lining  is  compressed,  enlarging  the  diame- 
ter slightly  and  also  causing  the  hose  to  straighten.  An  ex- 
periment on  300  feet  length  of  rubber  lined  hose  with  a 
guage  pressure  of  156  Ibs  per  square  inch  at  the  hydrant  end, 
showed  a  pressure  of  95  pounds  at  the  butt  of  the  nozzle,  or  a 
loss  by  friction  of  61  Ibs  in  300  feet  of  hose  while  the  dis- 
charge was  239  gallons  per  minute.  This  gives  the  formula 

P'=.00000356Xq*XZ 

The  difference  in  the  value  of  the  coefficient  for  very  low 
and  very  high  pressures  is  so  slight  as  to  be  of  no  practical  im- 
portance. It  will  be  understood  that  the  above  formula  does 
not  apply  to  leather  hose,  nor  to  any  other  than  2%  inch  rubber 
lined  hose.  The  coefficient  is  in  its  direct  form,  and  conse- 
quently applies  only  to  the  diameter  for  which  it  was  deter- 
mined. 


SULLIVAN'S  NEW  HYDRAULICS.  139 

36—  Pressure  Required  at  Hydrant  or  Steamer  to  Force 
the  Discharge  of  a  Given  Quantity  in  Gallons  per  Min- 
ute. As  the  hoee  we  are  considering  was  partially  throttled 
by  the  one  inch  smooth  nozzle  at  discharge,  the  total  pressure 
was  not  all  neutralized  by  resistance  nor  converted  into  ve- 
locity, but  a  large  portion  of  it  remained  to  balance  the  fric 
tion  in  the  nozzle  and  to  generate  the  velocity  through  the 
nozzle.  Therefore,  in  order  to  ascertain  the  value  of  the  co- 
efficient of  velocity  m,  we  must  taks  P=P'+Pv  only,  for  the 
hose,  (not  the  nozzle). 

To  do  this,  we  must  first  find  the  value  of  Pv,  or  the  amount 
of  pressure  which  generates  the  given  velocity  in  the  2^ 
inch  hose.  The  quantity  passing  through  the  hose  was  239 
gallons  per  minute. 

This  is  equal  .5311  cubic  feet  per  second.  The  area  of 
the  hose  is  =.0341  square  feet.  The  velocity  in  feet  per  sec- 
ond through  the  hose  was  therefore 

cubic  feet       .5311 
v=—  ra  --  :034T=15'57  feet' 

The  pressure  causing  this  velocity  was 
Pv=—  o  -  =1.6337  Ibs  per  square  inch. 
Hence,  P=rP'+Pv=61+1.6337=62.&34  Ibs. 


p  =  -y-  xq*X  I  ^'Xq'X  l  =Q00003656Xq1X  l. 

Therefore  the  total  pressure  at  hydrant  or  steamer  that 
is  required  to  force  the  discharge  of  a  given  number  of  gal- 
lons per  minute,  (q)  through  any  length  in  feet  of  2J^  inch 
rubber  lined  hose,  will  be 

P=.000003G56Xq8  X  1. 

This  does  not  include  the  pressure  required  to  balance 
the  friction  in  the  nozzle,  nor  to  lift  the  weight  of  the  water 


140  SULLIVAN'S  NEW  HYDRAULICS. 

when  the  nozzle  end  of  the  hoze  is  elevated.  This  value  of 
P  is  that  which  is  required  to  balance  the  friction  in  the 
hose  (not  the  nozzle)  and  to  generate  the  velocity  of  flow  in 
the  hose.  If  the  discharge  end  of  the  hose  is  elevated,  then 
sufficient  additional  pressure  must  be  added  to  the  above 
value  of  P  to  raise  the  weight  of  the  given  quantity  to  the 
given  height. 

The  pressure  lost  by  friction  in   2^  inch   leather   hose  is 

P'=.0000067464Xq2XZ 

q=gallons  discharged  per  minute. 

Z=length  in  feet  of  hose. 

From  this  value  of  the  coefficient  as  compared  with  the 
value  of  the  coefficient  for  rubber  lined  hose,  it  is  seen  that 
the  friction  loss  in  leather  hose  is  nearly  double  that  in  rub 
ber  hose.  For  this  reason  leather  hose  has  fallen  into  disuse 
and  will  therefore  not  be  discussed  further. 

37.— Loss    By    Friction    In    Brass     Fire    Nozzles.— 

In  conical  pipes  or  nozzles  which  converge  from  a  larger 
to  a  smaller  diameter,  the  velocity  is  inversely  as  the  con- 
stantly changing  area  and  the  resistance  is  inversely  as  ^/d3. 
The  velocity  and  resistance  are  therefore  different  at  each 
successive  point  along  the  length  of  such  convergent  pipe  or 
nozzle.  The  velocity  is  greatest  in  the  portion  having  the 
least  diameter  and  least  in  the  greatest  diameter.  If  we  take 
the  mean  of  all  the  varying  velocities  in  such  convergent  noz- 
zle, it  will  be  found  that  this  mean  is  very  much  greater 
than  the  mean  velocity  through  a  pipe  of  uniform  diameter 
which  uniform  diameter  is  equal  to  the  mean  or  averag-e 
diameter  of  the  convergent  nozzle.  It  is  therefore  evident 
that  the  friction  in  the  nozzle  will  greatly  exceed  that  in  the 
uniform  diameter. 

From  the  results  of  many  experiments  with  very  small 
nozzles  and  large  nozzles  of  cast  iron  from  8  to  12  feet  in 
length,  the  writer  has  discovered  that  the  friction  in  a  nozzle 
or  convergent  pipe  is  nine  times  as  great  as  in  a  pipe  of  uni- 
form diameter  which  uniform  diameter  equals  the  mean  di- 
ameter of  the  convergent  pipe,  both  being  of  the  same  mate- 


SULLIVAN'S  NEW  HYDRAULICS.  141 

rial  and  same  length,  and  discharging  equal  quantities  of 
water  in  equal  times. 

The  coefficient  of  resistance  n,  for  smooth  brass  in  terms 
of  head  and  diameter  in  feet  is  n=.0003268,  or  nearly  the 
same  as  for  asphaltum  coated  pipes.  A  smooth  brass  tire 
nozzle  18  inches  in  length  and  converging  from  2^  inches 
inside  diameter  at  the  butt  to  a  diameter  of  one  inch  at  dis- 
charge, discharged  .17134566  cubic  feet  per  second  when  the 
guage  pressure  at  the  buit  of  the  nozzle  was  10  pounds  per 
square  inch.  As  the  velocity  pressure  is  parallel  to  the  walls 
of  the  pipe,  it  is  not  shown  by  a  pressure  guage.  In  order  to 
find  the  friction  loss  in  the  nozzle  we  must  find  the  total 
pressure  at  the  butt  of  the  nozzle  and  then  find  the  pressure 
which  causes  the  velocity  of  final  discharge  from  the  nozzle. 
The  difference  between  the  total  pressure  at  the  butt  of  the 
nozzle  and  the  pressure  due  to  the  velocity  of  discharge  from 
the  nozzle,  is  evidently  equivalent  to  the  pressure  lost  by 
friction  in  the  nozzle. 

The  pressure  causing  the  velocity  in  the  hose  at  the  butt 
of  the  nozzle  is  to  be  found  and  added  to  the  guage  pressure 
at  the  butt  of  the  nozzle.  The  velocity  in  the  hose  while 

q  _  .17135 
discharging  .17135  cubic  feet  per  second   was  v=— 

—5.0248.    The  pressure  causing  this  velocity  is  Pv= 


=.17  Ib.  Add  this  to  guage  pressure  at  butt  of  nozzle  and  the 
total  pressure  at  the  butt  is  P=10.17  Ibs.  The  area  in  square 
feet  of  the  one  inch  discharge  of  the  nozzle  is  .0055.  Conse- 
quently the  final  velocity  of  discharge  from  this  one  inch 

nozzle   was    v  =— =~~0055 — =31.20  feet  per  second.     The 

pressure  causing  this  final  velocity  of  discharge  from  the  noz- 

va~y  4.34. 
zlewasPv=    2    '  ' — =6.55  Ibe.     The   pressure    lost  in  the 

nozzle  b^  friction  was  therefore  10.17— 6.55=362  Ibs,  or  8.34 
feet  head  while  the  discharge  was  .17135  cubic  feet  per  sec- 


142  SULLIVAN'S  NEW  HYDRAULICS. 

ond.  The  average  or  mean  diameter  of  this  nozzle  was  .1458 
foot,  and  the  area  of  this  mean  diameter  was  .0167  square 
foot.  Hence  the  velocity  through  the  mean  diameter  while 

discharging  ,17i35  cubic  feet  per  second  was  v—^—=   nig7~ 

=10.26  feet.  The  coefficient  of  resistance  of  a  smooth  brass 
pipe  of  uniform  diameter  is  n=.0003268.  Hence  the  loss  of 
head  in  feet  in  a  smooth  brass  pipe  of  uniform  diameter 
equal  to  the  mean  diameter  of  this  nozzle,  and  of  equal  length 

.0003268  .0003268X1.50X105.2676 

wouldbeh'=-73f-XlXV=  .0557 

.9267  feet.  This  is  equal  to  only  one  ninth  part  of  the  actual 
loss  in  the  convergent  nozzle.  Hence  in  a  formula  for  friction 
loss  in  a  conical  or  convergent  pipe  or  nozzle  we  must  take 
the  square  of  three  times  the  velocity  through  the  mean  di- 
ameter (3Xv)a  or  (3Xq)2=9v8  or  9q8,  or  we  must  find  the  co- 
efficient of  friction  n  in  terms  of  quantity  or  velocity  and 
multiply  by  9  for  a  convergent  nozzle  or  pipe,  or  we  must 
consider  the  nozzle  as  a  pipe  of  uniform  diameter  and  as  be- 
ing 9  times  as  long  as  the  nozzle.  If  we  consider  it  as  a  uni- 
form diameter  then  that  diameter  must  be  equal  to  the  aver- 
age diameter  of  the  nozzle  or  conical  pipe  and  nine  times  as 
long. 

Hence  the  general  formula  for  loss  of    head    in  feet  by 
friction  in  conical  pipes,  reducers  and  nozzles  will  be 


9v» 

In  which 

d=mean  or  average  diameter  of  the  convergent  pipe. 
v=velocity  in  feet  per  second  in  the  mean  diameter. 
n=coefficient  of  friction  in  same  terms  as  d. 
The  above  value  of  n  is  iu  terms  of  head  and   diameter  in 
feet. 


SULLIVAN'S  NEW  HYDRAULICS. 


143 


In  a  nozzle  of  given  length  and  form  the  loss  by  friction 
will  vary  directly  as  the  head  or  pressure  at  the  butt  of  the 
nozzle,  or  directly  as  Vs  or  q8.  Hence  a  constant  multiplier 
may  be  determined  for  each  form  and  length  of  nozzle,  by 
which  the  loss  for  any  discharge,  head  or  pressure  may  at 
once  be  found.  For  example,  if  the  formula  is  in  terms  of  di- 
ameter in  feet,  pressure  in  Ibs.  per  square  inch  at  butt  of  noz- 
zle (guage  pressure  +Pv.)  and  v8,  then 
.0001418X  I X9  vs 


P  ': 


-,  for  brass  smooth  (not  ring)  nozzles 


From  which  the  following  table  of  constants  was  calcu- 
lated: 

TABLE  No.  20. 

Table  of  multipliers  for  finding  pressure  lost  by  friction 
in  brass  smooth  (not  ring)  fire  nozzles.  For  any  head  or  total 
pressure. 


Length  of 
Nozzle 
Inches 

Diameter 
at  Butt 
Inches 

Diameter 
at  Discharge 
Inches 

Lbs.  pressure  lost 
equal   total   pres- 
sure at  butt  mul- 
tiplied by  the  dec- 
imal below: 

18.000 
12.000 
3.500 
18.000 
12.000 
3.204 
18.000 
12.000 
2.9125 

2  1-2 
2  1-2 
2  1-2 
2  1-2 
2  1-2 
2  1-2 
2  1-2 
21-2 
2  1-2 

!l-8 
.1-8 
1-8 
.1-4 
.1-4 
.14 

.356 
.2373 
.06922 
.49 
.325 
.087 
.474 
.316 
.0767 

These  multipliers  exhibit  the  relative  efficiency  of  fire 
nozzles  of  different  lengths  and  forms,  and  show  the  import- 
ance of  making  nozzles  and  reducers  of  short  length.  For 
the  least  loss  and  greatest  efficiency  the  rate  of  convergence  in 
a  reducer  or  nozzle  should  be  one  inch  in  a  length  of  2.33  in- 
ches,which  will  conform  to  the  shape  of  the  contracted  vein  or 
vena  contracta.  (See  §  80.) 

If  we  wish  to  determine  the  direct  coefficient  for  a  given 
length  and  form  of  nozzle  in  terms  of  gallons  discharged  per 
minute  and  pressure  in  Ibs.  per  square  inch,  take  the  experi- 


144  SULLIVAN'S  NEW  HYDRAULICS. 

mental  data  already  given,  for  example,  and  we  have 
1=1.5    feet=18  inches 
d=.1458  feet=mean  diameter^2'5+1=1.75  inches 

P'  =10.17—  6.55=3.62  Ibs. 

Discharge=:.17185  cubic  feet  per  second=77  gallons   per 
minute 

P'd_  3.62X1.75  _ 


Z=feet,  and  demean  diameter  in  inches 
q=gallons  per  minute. 

CAUTION.  This  last  formula  is  in  the  direct  form,  and 
will  apply  only  to  the  given  nozzle  for  which 
it  was  determined.  If  the  direct  coefficient, 
.000407,  be  multiplied  by  the  constant  length 
in  feet  I  of  the  given  nozzle,  then  .000407X1.5 
=0006105,  and  the  loss  of  pressure  by  friction 
in  this  given  form  and  length  of  nozzle  for 
any  discharge  in  gallons  per  minute  is 

P'=.0006105Xq2. 

A  direct  constant  may  be  found  in  the  same  manner  for 
each  length  and  form  of  nozzle. 

It  is  interesting  to  compare  the  values  of  n  for  different 
materials  when  the  unit  values  of  n  are  all  in  the  same  terms. 
Thus  u=0001418  for  smooth  brass.  n.=0000754  for  rubber. 
These  are  the  unit  values  of  n  in  terms  of  P'  and  diameter  in 
feet,  showing  that  rubber  offers  less  resistance  to  flow  than 
smooth  brass  or  asphaltum  coatings. 

In  a  constant  diameter  of  pipe,  or  in  a  constant  length 
and  form  of  nozzle,  the  friction  will  increase  or  decrease  di- 
rectly as  the  pressure  or  head.  Hence  if  a  total  pressure  at 
the  butt  of  the  nozzleiof  10.17  Ibs  will  cause  a  loss  by  friction 
of  3.62  Ibs  in  the  given  nozzle,  then  a  total  pressure  of  one  Ib. 

at  the  butt  would  cause  a  friction  loss=  3'62  =.356  Ib.,  and 

10.17 


SULLIVAN'S  NEW  HYDRAULICS,  145 

any  other  total  presssure  at  the  butt  would  cause  a  loss  of 
P'=PX-356,  for  the  given  nozzle. 

If  a  slope  S— .0004  in  a  cast  iron  pipe  one  foot  diameter 
will  cause  a  loss  of  .0003938  foot  head  per  foot  length  of  pipe, 
then  the  loss  of  head  for  any  other  slope  of  a  one  foot  pipe 

would  be=^.—3?3?_=.9845XS.      And    so    of  any  other  con- 

.0004: 

stant  diameter  or  form  of  pipe  or  nozzle. 

As  friction  increases  as  the  square  of  the  quantity  dis- 
charged, if  the  loss  by  friction  in  the  nozzle  is  3.62  Ibs.  while 
it  is  discharging  17  gallons  per  minute,  the  loss  for  a  dis- 

o  £»Q  q  £»o 

charge  of  one  gallon  per  minute  would  be  -         — 


(77)2         5929 

.0006105  Ibs.,  and  for  any  other  discharge  in  gallons  per  min- 
ute it  would  be  =  .0006105X(gallons)2. 

If  the  loss  of  head  in  feet  by  friction  in  each  foot  length 
of  a  12  inch  diameter  cast  iron  pipe  is  .0003938  foot  while  the 
pipe  is  discharging  .7854  cubic  foot  per  second,  thon  the  loss 
fora  discharge  of  one  cubic  foot  per  second  in  such  diameter 

will  be  =  •00°g^88  =.0006384  foot  head  per  foot    length  and 

for  any  greater  or  less  discharge  in  cubic  feet  per  second  the 
loss  of  head  per  foot  length  will  be 

h"  =  .0006384X (cubic  feet  per  second)8. 
Hence  it  is  a  simple  matter  to  find  the  proper  constant  in 
terms  of  head,   pressure,  velocity,  slope  or  quantity  for  any 
given  form  of  nozzle  or  for  any  given  diameter. 

38— Friction  In  Ring  Fire  Nozzles.— On  account  of  the 
abrupt  shoulder  or  offset  caused  by  the  sudden  contraction 
of  the  diameter  by  the  ring  in  what  is  termed  a  ring  fire  noz- 
zle, very  serious  reactions  and  eddy  effects  occur  in  such  noz- 
zles, and  the  loss  of  head  or  pressure  thus  caused  is  very 
great.  In  an  experiment  with  a  ring  nozzle  of  brass,  18  inches 
in  total  length,  with  a  butt  diameter  of  2}£  inches  and  a  ring 
one  inch  diameter,  and  a  total  pressure  at  the  butt  equal  to 
23.237  feet  head,  the  nozzle  discharged  .  13333  cubic  feet  per 
second.  The  velocity  through  the  one  inch  ring  was  there- 


146  SULLIVAN'S  NEW  HYDRAULICS. 

fore  v=  q  —  -13333  =24.243  feet  per  second.  The  head 
due  to  this  final  velocity  of  discharge  from  the  nozzle  was 

Hv=-^-=  587>72  =9.126  feet  head. 

2g         64.4 

Deducting  this  from  the  total  head  at  the  butt  of  the 
nozzle,  and  the  friction  loss  in  the  nozzle  was  23,237—9.126= 
14.111  feet  head  or  more  than  half  the  total  pressure  at  the 
butt  of  the  nozzle. 

39—  Hydraulic  Giants,  Cast  Iron  Nozzles  for  Power 
Mains,  Reducers,  and  Conical  Pipes  In  General.—  The 

writer  has  made  many  experiments  on  cast  iron  giants  or  con- 
vergent pipes  of  various  dimensions  and  under  heads  of  20 
to  600  feet  at  the  base  of  the  giant.  The  results  of  these  ex- 
periments confirm  the  correctness  of  the  general  formula 
heretofore  given  for  finding  the  loss  by  friction  in  nozzles, 
reducers  and  convergent  pipes—  that  is  to  say,  the  friution 
in  a  cast  iron  giant  or  convergent  pipe,  will  be  nine  times  as 
great  for  the  same  discharge  as  it  would  be  in  a  uniform  di- 
ameter equal  to  the  mean  diameter  of  the  giant,  reducer  or 
convergent  pipe.  Hence  the  general  formula  for  head  in 
feet  lost  by  friction  in  such  giant  or  convergent  pipe  is 

*xz  ..............  (95) 


In  this  formula 

d=the  mean  or  average  diameter  of  the  giant. 

v=velocity  in  the  mean  diameter  in  feet  per  second. 

n=the  usual  coefficient  of  resistance  for  the  class  of  cast 
iron  or  other  material. 

I  ^length  in  feet  of  giant. 

If  d  is  taken  in  inches  then  n  must  also  be  in  the  same 
terms. 

Cast  iron  giants  for  discharging  water  upon  impulse 
water  wheels  are  required  to  be  of  the  best  metal  and  without 
flaws.  They  are  usually  under  high  pressure  and  the  veloci- 
ties through  them  are  terrific.  Hence  they  are  scoured  and 


SULLIVAN'S  NEW  HYDRAULICS.  U7 

kept  clean  BO  the  coefficiennt  will  not  increase  after  long  use, 
unless  the  water  contains  sand  or  gritty  matter  which  cuts 
the  pipe  walls  and  roughens  them. 

For  this  very  dense,  smooth  cast  iron,  as  usually  found 
in  such  nozzles,  n=  .0003623  in  terms  of  diameter  in  feet. 
Using  the  value  of  the  mean  diameter  in  feet  of  the  cast  iron 
nozzle  and  the  velocity  v,  through  the  mean  diameter,  and 
the  general  formula  for  friction  in  such  cost  iron  giants  is 


The  coefficient  n  may  be  determined  in  terms  of  quantity 
discharged  per  second  or  per  minute  so  that  the  discharge 
will  correspond  with  a  given  loss  of  head.  In  an  experiment 
with  a  cast  iron  giant  8  feet  in  length  and  converging  from 
a  diameter  of  fifteen  inches  at  the  base  to  a  diameter  of  3 
inches  at  discharge,  the  loss  of  head  in  feet  by  friction  while 
the  discharge  was  8.845  cubic  feet  per  second  was  16.10  feet 
head. 

The  mean  diameter  = „ — ==  9  inches  =.75  foot. 

Area  of  mean  diameter=(.75)a  X-7854=.4418  square  foot. 

Velocity  through  mean  diameter  —  ^  =  ^  g  =20022  ft. 

per  second. 

Velocity  in  15-inch  diameter  =7.209  feet  per  second. 

Velocity  of  discharge  in  3-inch  diameter  =180.16  feet  per 
second. 

The  mean  of  the  velocities  in  all  diameters  =93.68  feet 
per  second. 

Using  the  value  of  n  applicable  to  this  class  of  dense 
cast  iron  and  n=.0003623  in  terms  of  head  and  diameter  in 
feet.  Then, 

.0003623X9  .0032607 

b"  = ^3 Xv8  1=       b  X(20.022)  2X8  =16,10 


148 


SULLIVAN'S  NEW  HYDRAULICS. 


feet  head  lost  by  friction.  This  corresponds  exactly  with 
the  actual  result. 

In  this  given  nozzle  therefore,  as  the  loss  of  head  in  feet 
by  friction  was  16.10  feet  while  the  discharge  was  8.845  cubic 
feet  per  second,  the  loss  by  friction  for  a  discharge  of  one  cu- 
bic foot  per  second  would  be  h'  =  16-1°  =.2058  foot,  and  for 

(O.OiO)8 

any  other  discharge  in  cubic  feet  per  second  it  would  be  h"= 
.2058Xq*.  Here  qs=cubic  feet  per  second  discharged.  A  con- 
stant for  any  other  length  and  mean  diameter  of  nozzle  or 
conical  pipe  may  be  readily  found  in  the  same  manner  in  any 
terms  desired.  (See  Table  No.  25,  §  56,) 

40.— Table  of  Multipliers  for  Determining  the  Loss  of 
Head  in  Feet  by  Friction  in  Clean  Cast  Iron  Nozzles  of 
Given  Dimensions. 

TABLE  No.  21,  (See  Table  No.  26.) 


Head  in   feet  lost 

Length 
in  Feet  of 
Nozzle 

Greatest 
Diameter 
Inches 

Least 
Diameter 
Inches 

n  nozzle  equals  ef- 
'ective      head     at 
aase  of  nozzle  mul- 
tiplied by  the  dec- 

imal below 

8 

20 

5 

.0415 

8 

20 

4 

.02113 

8 

20 

3 

.00823 

8 

18 

4 

.0324 

8 

18 

3 

.012033 

8 

18 

2 

.003815 

8 

15 

3 

.031 

8 

15 

2K 

.01768 

8 

15 

2 

.0086 

8 

14 

2 

:<>iix5 

8 

12 

1*6 

.0098 

8 

12 

2 

.0244 

8 

12 

1 

.002394 

8 

10 

1 

.0058654 

42:  -The  Total  Head  fn  Feet  H,  or  the  Slope  Required 
to  Cause  the  Discharge  of  a  Given  Quantity  in  Cubic  Feet 
Per  Second  in  Ordinary  Cast  Iron  Pipes. 

By  formula  (30)  the  slope  required  to  cause  the  discharge 


SULLIVAN'S  NEW  HYDRAULICS.  149 

of  a  given  number  of  cubic  feet  per  second,  is 

m  q8  m  q2  q* 

.616853-j/d1 1 .616853       -j/o11  ~  "''^y'd11 

The  total  head  in  feet  in  the  length  in  feet  I,  required  to 
cause  a  given  diameter  of  common  cast  iron  pipe  to  discharge 
a  given  quantity  q,  in  cubic  feet  per  second,  is 

H=     ^g^    X-^nX  I  =.000648452X-^nX  I 

In  these  formulas  d   is  expressed  in   feet.    In  table  No. 
18  the  values  of  y'd11  are  given. 

43— The  Slope  or  Total  Head  in  Feet  Being  Given,  to 
Find  the  Diameter  in  Feet  of  Common  Cast  Iron  Pipe 
Required  to  Discharge  a  Given  Quantity  in  Cubic  Feet  per 
Second. 

From  the  above  formula,  S=.000648452X-^T- 
Whence, 


,000648452Xqs 

•d"=  --  s  ~ 

,qil_(.000648452)2Xq4 

—          - 


i2632 


=n/;jF 


Or  in  terms  of  total  head  in  feet,  for  the  given  value  of 
m. 


44.—  Head  in  Feet  Lost  by  Friction  In  Different  Diame- 
ters of  Clean,  Ordinary  Cast  Iron  Pipe  While  Discharging 
Given  Quantities  in  Cubic  Feet  Per  Second. 

By  formula  (32)  the  head  in  feet  lost  by  friction  for  a 
given  discharge  is 

h"==.616863v/d"  ...............................  (32) 


150  SULLIVAN'S  NEW  HYDRAULICS. 

The  value  of  n  in  terms  of  diameter  in  feet   for  ordinary 
clean  cast  iron  pipe  is  n  =  . 0003938.    Hence. . . . 


.0006384 
h"= — I/dTrXq^XJ-    It  is  convenient  to  have  the  loss  of 

head  per  100  feet  length  of  pipe,  and  therefore  we  may  make 
1=100  feet  as  a  constant.  The  loss  of  head  in  feet  by  friction 
in  each  100  feet  length  of  pipe  for  any  given  discharge  in 
cubic  feet  per  second  will  then  be 

h"=  ' /Jn  Xq*  •  Now  if  we  take  the  quotent   of-'  /^n" 

for  each  diameter  of  pipe  in  feet,  the  result  will  be  a  constant 
for  that  diameter  in  feet,  and  when  such  constant  is  multi- 
plied by  the  square  of  the  discharge  in  cubic  feet  per  second, 
the  product  will  equal  the  loss  of  head  in  feet  per  100  feet 
length  of  that  diameter  for  the  given  discharge. 

To  facilitate  such  calculations  the  following  table  of 
such  constants  is  given. 

45— Table  of  Multipliers  for  Determining  the  Loss  of 
Head  in  Feet  by  Friction  Per  100  Feet  Length  of  Ordi- 
nary Clean  Cast  Iron  Pipe  for  a  Given  Discharge  In  Cubic 
Feet  Per  Second.— 

TABLE  No.  22. 

Multiply  the  constant  which  corresponds  with  the  given 
diameter  on  same  line  in  the  table  by  the  square  of  the  dis- 
charge in  cubic  feet  per  second  (q8).  The  result  will  be  the 


SULLIVAN'S  NEW  HYDRAULICS. 


151 


loss  of  head  in  feet  per  100  feet   length  of  pipe   for  that  dis- 
charge. 


Diam- 

Inch's 

Diam. 
Feet 

.06384 

Diam 
Inch's 

Diam. 
Feet.  v 

.06384 

7"^*^- 

"~  Constant 

V/d11 
Constant 

2 

.1667 

1214.611000 

23 

1.917 

0.00178125 

3 

.25 

130.737000 

24 

2.000 

0.00141083 

4 

.3333 

26.880000 

25 

2.0-3 

0.00112791 

5 

.4167 

7.871700 

26 

2.166 

O.Q0090979 

6 

.5 

2.900000 

27 

2.250 

0.00073800 

7 

.5833 

1.236000 

28 

2.333 

0.00060483 

8 

.6667 

0.594000 

29 

2.416 

0.00050000 

9 

.75 

0.310657 

30 

2.500 

0.00041347 

10 

.8333 

0.174045 

31 

2.584 

0.000344708 

11 

.9167 

0.10:3000 

32 

2.666 

0.000290000 

12 

1.000 

0.0o384 

33 

2.750 

0.000244785 

13 

1.083 

0.041187 

34 

2.834 

0.000207407 

14 

1.167 

0.027305 

35 

2.916 

0.000177333 

15 

1.250 

0.0187104 

36 

3.000 

0.000151675 

16 

1.333 

0.0131385 

38 

3.166 

0.000112800 

17 

1.417 

0.0093>>23 

40 

3.333 

0.000085018 

18 

1.500 

0.0(W8(i3-0 

42 

3.500 

0.000064970 

19 

1.583 

0.00510310 

44 

3.666 

0.000050347 

20 

1.667 

0.  003*41  l.V> 

48 

4.000 

0.000*311714 

21 

1.750 

0.00294000 

54 

4.500 

0.0000163100 

22 

1.833 

0.00227919 

60 

5.000 

0.00000900415 

For  convenience  in  referring  to  the  table  No.  22,  the  di- 
ameters are  given  first  in  inches  and  then  in  feet.  The  total 
head  per  100  feet  length  ia  equal  the  loss  of  head  per  100 
feet  divided  by  .9815,  provided  the  diameter  is  constant  and 

the  discharge  is  free,  or  H=h"Xl-01573=  9g45  . 


46.—  Head  in  Feet  Lost  by  Friction  In  Asphaltum 
Coated  Pipes  While  Discharging  a  Given  Quantity  in  Cubic 
Feet  Per  Second.— it  we  refer  to  the  table  of  values  of  m  as 
developed  in  Group  No.  2  from  the  experiments  of  Hamilton 
Smith  Jr.,  and  of  D'Arey  and  Bazin,  on  this  class  of  pipe,  it 
will  be  seon'that  the  value  of  m,  the  coefficient  of  velocity, 
varies  from  m=.00028  to  m=.0003432  in  the  experiments  of 
Smiti.,  and  from  m=  000271  to  m=.000289  in  the  experiments 
of  D'Arcy.  Smith's  experiments  were  on  lap  seamed  riveted 


152  SULLIVAN'S  NEW  HYDRAULICS. 

pipes  with  Blip  joints  like  stove  pipe  joints.  D'Arcy's  experi- 
ments were  on  cast  iron  coated  pipes  which  were  free  of  rivet 
heads  and  seams,  but  which  were  in  shorter  lengths  and  re- 
quired more  joints.  The  coefficient  of  friction  n,  for  any  given 
class  of  perimeter,  is  always  equal  .9845  per  cent  of  the  value 
of  m  for  the  given  class  of  perimeter.  Hence  where  the  value 
of  m  is  known  for  any  class  of  perimeter,  the  value  of  n  for 

n 
that  class  is  n=mX-9845,  and  m=  .      As   it  is   prudent 


to  allow  for  errors  in  the  experimental  data  from  which  the 
above  values  of  m  were  deduced  and  also  for  inferior  quality 
of  the  coating,  and  future  deterioration  of  the  coating  and 
slight  deposits,  we  will  adopt  the  value  of  n=.  00032  in 
terms  of  diameter  and  head  in  feet.  This  should  be  a  safe 
and  reliable  value  of  n  for  either  riveted  pipe,  welded  pipe,  or 
cast  iron  pipe,  which  has  been  coated  with  asphaltum.  The 
coating  material  usually  covers  the  rivet  heads  and  fills  the 
longitudinal  offset  made  by  the  lap  of  the  plate.  Hence  there 
should  not  be  a  great  difference  in  the  value  of  n  or  m  for 
either  class  of  pipe  after  it  has  been  coated.  D'Arcy's  coeffi- 
cients are  usually  too  small  (that  is,  m  or  n,  which  makes  C 
too  high)  and  the  length  of  pipe  used  in  his  experiments  was 
rather  short.  The  experiments  of  Smith  are  considered  more 
reliable.  They  are  safer  to  use  in  practice  at  any  rats. 

TABLE  No.  23. 

Table  No.  23  is  based  on  the  same  principle  as  table  No. 
22,  and  its  use  is  fully  explained  in  §  44,  45. 

To  USE  TABLE  No.  23. 

To  find  the  loss  of  head  in  feet  per  100  feet  length  of  as- 
phaltum coated  pipe  for  any  given  discharge  in  cubic  feet  per 
second,  multiply  the  constant  in  3d  column  opposite  the  giv- 
en diameter  by  the  square  of  the  discharge  in  cubic  feet  per 
second,  qs. 


SULLIVAN'S  NEW  HYDRAULICS.  153 

.051884 


Head  in  feet  lost   per  100  feet  length,   h' 
d  is  in  feet. 


dll 


Diam- 

Diam- 

.051864 

Diam 

Diam 

.051864 

eter 
In. 

eter 
Feet. 

Constant 

eter 
In. 

eter 
Feet. 

Constant 

2 

.1667 

986.758 

23 

1.917 

0.00144710 

3 

.25 

106.213 

24 

2.000 

0.001146165 

4 

.3333; 

21  .  417 

25 

2.0^3 

0.000916325 

5 

.4167 

6.400 

26 

2.166 

0.000739120 

6 

.5 

2.347 

27 

2.25 

0.000599600 

7 

.5833 

1.006 

28 

2.333 

0.000491370 

8 

.6667 

0.482460 

29 

2.416 

0.000405200 

9 

.75 

0.252380 

30 

2.5 

0.000386000 

10 

.8333 

0.141400 

31 

2.584 

0.000280000 

11 

.9167 

0.083680 

32 

2.666 

0.000236852 

12 

1.000 

0.051864 

33 

2.75 

0.000199000 

13 

1.083 

0.033460 

34 

2.834 

0.000170000 

14 

.167 

0.022183 

35 

2.916 

0.000144070 

15 

.25 

0.015200 

36 

3.000 

0.000123222 

16 

.333 

0.0106738 

38 

3.166 

0.0000916325 

17 

.417 

0.0076270 

40 

3.333 

0.0000690690 

18 

.5 

0.00557617 

42 

3.500 

O.OOOOV27820 

19 

.583 

0.004145M) 

44 

3.666 

0.0003409000 

20 

.667 

0.00312060 

48 

4.000 

0.0000253241 

21 

.75 

0.00240000 

54 

4.500 

0.0000132509 

22 

.833 

0.00185160 

60 

5.000 

0.0000074314 

REMARK— The  value  of  n  used  in  above  taole  will  allow 
for  the  reduction  of  area  and  diameter  by  the  thickness  of 
the  coating,  BO  that  the  actual  diameter  before  it  is  coated 
may  be  used  without  any  allowance  for  thickness  of  the  coat. 

47.— To  Find  the  Quantity  Discharged  when  the  Loss 
of  Head  and  Diameter  are  Given.— The  quantity  in  cubic 
feet  per  second  which  is  being  discharged  by  any  diameter 
may  be  found  from  the  loss  of  head  as  indicated  by  pressure 
guages.  We  have  just  seen  that  the  loss  of  head  per  100  feet 
length  of  coated  pipe  for  a  given  discharge  in  cubic  feet  per 


second  is  h"  = 


051864 


Xq2.     By  transposing  in  this  equation 


we  have  the  formula  for  finding  the  quantity  discharged  in 
cubic  feet  per  second  from  the  amount  of  head  in  feet  lost  by 
friction  h",  thus, 

''whence'q=AJS&r 


154  SULLIVAN'S  NEW  HYDRAULICS. 

d=feet,  and  h"=head  in  feet  lost  per  100  feet  length  of  the 
pipe. 

A  similar  formula  for  cast  iron  pipe  may  be  deduced 
from  the  coefficient  values  given  in  §  44.  The  values  of  ^/d11 
will  be  found  in  table  No.  18.  For  ordinary  cast  iron  pipe, 

not  coated,  we  have  h"='06384   Xq*=loss  per  100  feet  length 
•j/d1 1 

of  pipe.  Hence  the  discharge  in  cubic  feet  per  second  cor- 
responding with  this  loss  of  head  in  feet  per  100  feet  length 
of  pipe  is 


48-  To  Find  the  Quantity  that  a  Given  Slope  will  Cause 
a  Given  Diameter  to  Discharge. 

The  slope  required  to  cause  the  discharge  of  a  given  quan- 
tity in  cubic   feet  per  second  is  S=- 
.00064845^ 

By  transposition  we  have 

/    Si/d11  / 

q=  I/    .0006^845  =  39.27 j/Syd"  =39.27 Vd11  X  v/S, 

H__  total  head  in  feet 

~   I        total  length  of  pipe  in  feet 
d=diameter  in  feet. 
q=cubic  feet  per  second  discharged. 
Table  No  18  gives  the  values  of  yd",  and  Table  No.  15 
gives  yS. 

49— To  Find  the  Total  Pressure  in  Pounds  Per  Square 
Inch  that  must  be  Exerted  by  a  Pump  Piston,  or  by  Other 
Means,  in  Order  to  Cause  a  Given  Diameter  of  Asphaltum 
Coated  Pipe  to  Discharge  a  Given  Quantity  in  Cubic  Feet 
Per  Second. 

By  formula  (44) 


SULLIVAN'S  NEW  HYDRAULICS.  155 


As  we  have  adopted  n=. 00032  as  the  safe  coefficient  of 
friction  in  terms  of  head  and  diameter  in  feet  for  asphaltum 
coated  pipes,  the  corresponding  value  of  the  coefficient  of  flow 

would  be  m  —  Q^  =.00032503,  in  terms  of  head  and   diame- 
ter in  feet. 

To  reduce  this  value  of  m  to  terms  of  pressure  in  pounds 
per  square  inch  and  diameter  in  feet,  it  is  simply  necessary  to 
divide  by  the  number  of  feet  head  required  to  cause  a  pres- 
sure of  one  pound  per  square  inch.  II=PX2.304,  and  P= 

H 
2.304  ' 

Therefore  if  P=l  pound  per  square  inch,  then  H— 1X2.- 
304=2.304  feet. 

00032503 

Hence,  m=~2~304 — =  0001il07)  *n  terms  of  ^  an(*  d  in 
feet. 

.00014107          q" 
Then,  from  formula  (44),  P=     t61685^X  ^/dl^^l= 

.0002287 

1/cin-Xq'xJ. 

The  total  pressure  to  be  exerted  by  the  pump  is  there- 
fore, 

.0002287 

P=Vd^Xq  Xl 

d  =  diameter  of  coated  pipe  in  feet. 

J=length  of  pipe  in  feet. 

q=cubic  feet  per  second  discharged. 

See  Table  No.  18  for  values  of  ^d*1. 

CAUTION:— It  is  assumed  in  the  above  formula  that  the 
pipe  is  laid  level,  or  that  there  is  no  differ- 
ence in  level  between  its  two  ends.  If  the 
pipeia  laid  on  a  declivity,  then  this  declivity 
would  supply  a  portion  of  the  head  or  pressure. 


156  SULLIVAN'S  NEW  HYDRAULICS. 

If  the  discharge  end  of  the  pipe  is  above  the 
pump,  then  additional  pressure  will  be  re 
quired  at  the  pump  sufficient  to  raise  the 
weight  of  the  given  number  of  cubic  feet  per 
second  to  a  height  in  feet  equal  to  the  differ- 
ence in  level  between  the  pump  and  th«  dis- 
charge end  of  the  pipe. 

50.— To  Find  the  Quantity  Discharged  From  the  Pres- 
sure. 

By  transposition  in  the  above  formulafor  P,  we  have 

coated  pi?6- 

P=toial  pressure  in  pounds  per  square  inch. 

d-diameter  in  feet  of  pipe. 

Z=length  of  pipe  in  feet. 

See  "Caution"  above.  For  value  of  ^/d11  see  Table  No 
18. 

For  cast  iron  pipe,  not  coated,  m=.0004  in  terms  of  head 
and  diameter  in  feet.  Hence  in  terms  of  P  and  d,  it  will  be 

m=^3Q|=.0001736Jl.  Therefore   P  — '    616853    X 


59.6 


51.  ~  Pounds  Pressure  Per  Square  Inch  Lost  by  Friction 
fora  Given  Discharge  In  Cubic  Feet  Per  Second. 

By  formula  (45)  the  pressure  lost  by  friction  for   a  given 
discharge  is 


As  we  have  just  found  the  values  of  m  in  terms  of  diam- 
eter in  feet  and  pressure  in  Ibs  per  square    inch,  for  coated 


SULLIVAN'S  NEW  HYDRAULICS.  157 

pipes  and  for  uncoated  cast  iron  pipes,  the  corresponding  val- 
ues of  n  will  be  n=mX-9845. 

Hence  for  asphaltum  coated  pipe  n=.00014107X.9845== 
.000138883415. 

For  cast  iron  pipe  not  coated,  n=. 00017361  lX-9845= 
.00017092. 

The  pressure  in  Ibs  per  square  inch  lost  by  friction  for  a 
given  discharge  in  cubic  feet  per  second  will  be,  for  coated 
pipe, 

.00017092         q*  .000225 

~  .616853    X7d^~><  /dH  Xq  X  '• 

And  for  cast  iron  pipe  not  coated, 

.000138883415  xx      q»  .000277 

P'= .616853      X7gn-X*=— ^TdTlXq'X* 

The  quantity  discharged  may  be  found  from  the  loss  of 
pressure  thus 


/    P'  -/d11 
q  =  V  .000225X  I    'f°r  C°ated  pipe* 


/  PVd11 
q  —  "\  OOQ977N/  /  '  'or  ca8*  iron  PiPe  no*  coated. 

See  table  No.  18  for  value  of  -/d'  1 . 

52.— Table  lor  finding  the  Slope  of  a  Cast  Iron  Pipe 
or  the  Total  Head,  in  Feet  Required  to  Cause  a  Given 
Discharge  in  Cubic  Feet  per  Second. 

The  quantity  discharged  by  a  constant  diameter  will  be 
directly  as  the  velocity  of  flow.  The  velocity  of  flow  will  be 
as  i/H  or  -j/S.  Hence  S  or  H  must  vary  as  vs  or  q*.  If  the 
slope  or  total  head  required  in  any  given  diameter  of  pipe, 
one  foot  in  length,  to  cause  a  discharge  of  one  cubic  foot  per 
second,  be  found,  then,  as  S  or  H  must  vary  as  q2  for  that 
given  diameter,  it  follows  that  the  slope  or  total  head  re- 
quired for  any  other  discharge  will  be  equal  to  the  slope  or 
head  which  causes  a  discharge  of  one  cubic  foot  per  second 


158  SULLIVAN'S  NEW  HYDRAULICS. 

multiplied  by  the  square  of  the   desired   discharge    in    cubic 
feet  per  second,  q*. 

If  the  required  slope  S  is  found,  then   the   total   head  in 

TT 

feet  for  any  given  length  in  feet  will  be  H=SX  M°r  S=  —  = 

rr 

total  head  required  per  foot  length.    '  =~g~;  H=SX  I- 
By  formula  (30), 

tn  Q  ^  Q  * 

TT  =  -00064845  X         n"'     —0004. 


Hence  the  slope,  or  the  total  head  in  feet  pei  foot  length 
of  any  given  diameter  of  ordinary  cast  iron  pipe,  not  coated, 
required  to  cause  the  discharge  of  one  cubic  foot  per  second 
will  be 


/jii     •     And  the  slope  required   to   cause   the  dis- 
charge of  any  greater  or  less  quantity  in  cubic  feet  will  be 
.     ,00061845 


In  which, 

d=diameter  of  pipe  in  feet. 
'q=cubicfeet  per  second. 

And  the  total  head  in  feet  required  in  any  given  length 
in  feet  of  pipe  will  be  H=SXf- 

TABLE  No.  24. 

To  find  the  slope  required  to  cause  any  given  diameter  in 
feet  of  uncoated  cast  iron  pipe  to  discharge  a  given  quantity 
in  cubic  feet  per  second:—  Rule.  —  Multiply  the  slope  in  the 
following  table  (No.  24)  which  is  opposite  the  .given  diameter, 
by  the  square  of  the  desired  discharge  in  cubic  feet  per  sec- 
ond, 

H=SXl. 


SULLIVAN'S  NEW  HYDRAULICS, 


159 


Diam 
eter 

r/dll 
froat 

Slope 
.00064845 

Diam- 
eter 

v/d*1 

Pftftt 

Slope 
.00064845 

Feet. 

r  eel 

v/d11 

Feet 

c  661 

^d" 

.1(567 

.00005256 

12.33734 

1.917 

35.84 

.0000180929 

.25 

.0004883 

1.32800 

2.000 

45.25 

.0000143300 

.3333 

.002375 

.27303 

2.083 

56.60 

.0000114567 

.4167 

.00811 

.07996 

2.166 

70.17 

.0000092410 

.5 

.Oi21 

.0293416 

2.25 

86.50 

.0000074:3653 

.5833 

.05157 

.0125740 

2.333 

105.55 

.00000614353 

.6667 

.1075 

.0060800 

2.416 

128.00 

.00000506600 

.75 

2055 

.0031555 

2.5 

154.40 

.0000042000 

.8333 

13668 

.0017680 

2.584 

185.20 

.0000035000 

.9167 

.6198 

.00104622 

2.666 

219.90 

.0000029500 

.000 

1.000 

.000o4845 

2.75 

260.80 

.00000248638 

.083 

1.55 

.0004184 

2.834 

307.80 

.00000210672 

.167 

2.338 

.00027735 

2.916 

360.00 

.00000180120 

.25 

3.412 

.00019000 

3.000 

420.90 

.00000154060 

.343 

4.859 

.0001334533 

3.166 

566.00 

.00000114570 

.417 

6.800 

.OOOoa-,:;--m 

3.333 

750.90 

.000000863563 

.5 

9.301 

.0000097180 

3.5 

982.60 

.000000660000 

.583 

12.51 

.0000518345 

3.666 

1268.00 

.000000511400 

1.667 

16.62 

.0000390000 

4.000 

2048.00 

.000000316621 

1.75 

21.71 

.00002'.iNV"7 

4.5 

3914.00 

.00000  '165674 

1.833 

28.01 

.0000231500 

5.000 

6979.00 

.000000092916 

See  Table  No.  16.  These  tables  apply  to  pipes  flowing 
full  bore  and  with  free  discharge.  q:q  ::i/S:v/S,  for  a  given 
diameter. 

S3.— Wooden  Stave  Pipes. — In  the  western  states,where 
irrigation  is  practiced  on  an  extensive  scale,  and  in  localities 
without  railway  facilities,  wooden  stave  pipe,  invented  by  Mr. 
J.  T.  Fanning,  and  described  in  his  "Treatise  on  Water  Sup- 
ply and  Hydraulic  Engineering"  page  439.  has  been  adopted 
in  many  instances  in  recent  years. 

In  the  very  dry  atmosphere  of  the  arid  west  these  pipes 
have  not  proven  satisfactory  in  many  cases  where  they  were 
laid  on  the  surface  or  without  sufficient  covering.  In  such 
cases  it  shrinks  and  warps  and  leaks  badly.  Where  properly 
covered  and  kept  constantly  full  of  water  it  has  been  quite 
satisfactory.  It  has  not  been  in  general  use  for  a  sufficient 
length  of  time  to  test  its  durability.  That  would,  of  course, 
depend  upon  the  kind  of  wood  used  in  manufacturing  the 
staves,  and  upon  whether  it  was  perfectly  seasoned  and 
sound.  If  perfectly  seasoned  and  treated  with  tar  oil  or  sul- 
phate of  copper,  it  should  be  very  durable.  The  quantity  of 
this  class  of  pipe  which  is  being  used  of  late  years  in  the 


160  SULLIVAN'S  NEW  HYDRAULICS. 

West  for  irrigation  purposes  in  caseB  where  there  is  only 
Email  pressure  to  be  sustained,  and  the  general  belief  that 
this  wooden  pipe  is  smoother  and  will  give  a  higher  dis- 
charge under  like  conditions  than  uncoated  iron  pipes,  de- 
mands that  it  be  given  some  notice  here. 

54.  —  Coefficients  of  Flow  in  Wooden  Stave  Pipes 
Compared  with  the  Coefficients  of  Pipes  of  other  Material. 

By  referring  to  the  coefficient  values  developed  from  the 
data  of  D'Arcy  and  Bazin,  (See  group  No  5),  it  will  be  seen 
that  the  average  value  of  m  for  wooden  conduits  made  of 
closely  jointed, planed  poplar  lumber  is  m  =  . 000060  in  terms  of 
hydraulic  mean  radius  in  feet  and  head  in  feet.  The  rectan- 
gular wooden  conduits  used  in  these  experiments  did  not 
contain  the  great  number  of  joints  which  are  necessary  in 
forming  a  circular  conduit  of  wooden  staves.  It  is  fair  to  as- 
sume then  that  the  circular  wooden  conduit  built  up  of  narrow 
staves  with  its  many  joints  would  not  present  a  more  uniform 
surface  to  the  flow  than  the  rectangular  conduit  or  flume  of 
planed,  well  jointed  hard  wood. 

The  nature  of  the  wood  of  which  the  staves  are  made  as 
to  density  and  freedom  from  knots,  will  undoubtedly  affect 
the  value  of  the  coefficient.  It  appears  from  the  great  num- 
ber of  experiments  by  D'Arcy  and  Bazin  on  such  conduits 
(only  a  few  of  which  were  quoted  in  Group  No.  5)  that  m= 
.00006  is  about  the  average  value  of  the  coefficient  in  terms  of 
r  in  feet. 

If  this  value  of  m  be  reduced  to  terms  of  diameter  in  feet, 
we  have  for  well  jointed,  planed  hard  wood  conduits,  m= 

.00006X8=.00048,  or  C=^~-  =45.64,  in    terms  of  head  and 

diameter  in  feet.  For  average  cast  iron  pipe,  not  coated,  m  — 
.00040  and  0=50.00.  For  asphaltum  coated  riveted  pipes,  0= 
56.CO.  For  pipes  lined  with  mortar  composed  of  two-thirds 
cement  to  one-third  sand,  0=48.50.  It  is  therefore  apparent 
that  the  wooden  pipe  offers  much  greater  resistance  to  flow 
than  either  of  the  others,  and  will  therefore  require  a  greater 
diameter  for  an  equal  discharge. 


SULLIVAN'S  NEW  HYDRAULICS.  161 

The  slope  required  in  a  wooden  pipe  in  order  to  cause  it 
to  discharge  a  given  quantity  in  cubic  feet  per  second  would 
be 

»       "    v    q'       .MOM        q*    _  -0007781  v... 

~  .616853X  v/d1*  T  .616853  x  v/d11  "~  i/d11  xq  ' 
and  qiqii/Si^S. 

And  the  diameter  in  feet  required  to  discharge  a  given 
quantity  for  a  given  slope  will  be 


55.— Earthenware  Or  Vitrified  Pipe  —  This  class  of 
pipe  is  made  in  very  short  lengths  and  consequently  requires 
many  joints.  It  is  subject  to  unequal  settlement  and  leaks 
unless  very  great  care  is  taken  to  secure  a  firm  bearing  or 
foundation  upon  which  to  lay  the  pipe.  It  also  requires  care 
and  experience  to  make  and  properly  cement  the  joints.  If 
the  pipe  is  made  of  clay  containing  a  high  percentage  of 
aluminum  and  is  thoroughly  glazed  and  p roperly  laid  and 
very  carefully  jointed,  it  develops  a  coefficient  m=.00036  or  C 
—52.70,  in  terms  of  diameter  in  feet  and  slope  or  head  in  feet. 
It  therefore  offers  less  resistance  to  flow  than  very  smooth, 
dense,  clean  cast  iron  pipe,  provided  all  the  above  conditions 
as  to  laying  and  jointing  are  complied  with.  As  these  con- 
ditions are  scarcely  ever  fulfilled,  it  is  not  prudent  to  depend 
upon  a  greater  discharge  from  such  pipe  than  from  ordinary 
clean  cast  iron  pipe.  Hence  all  the  tables  heretofore  given 
for  cast  iron  pipe  may  be  adopted  as  applying  also  to  earth- 
enware glazed  pipe.  This  class  of  pipe  is  very  extensively  used 
for  house  drains,  small  sewers,  land  drains  and  irrigation  pur- 
purposes,  and  in  other  rough  work  where  great  care  and 
thorough  workmanship  are  not  usually  exercised.  Hence  it 
is  not  safe  to  take  the  value  of  C  greater  than  C=50,  or  m  = 
.0001  in  terms  of  head  or  elope  and  diameter  in  feet.  For 
small  sewers,  not  exceeding  about  18  inches  diameter,  this 
class  of  pipe  serves  well.  The  flow  of  sewage  is  probably  not 
eo  great  as  that  of  clear  water  because  of  the  suspended,  solid 


162 


SULLIVAN'S  NEW  HYDRAULICS. 


matter  that  it  carries.  The  value  of  C  for  a  sewer  would 
therefore  rot  be  quite  BO  great  as  for  pure  water  flowing  in 
the  samo  class  of  pipe  or  conduit.  C=50,  should  be  a  safe 
value  for  fairly  well  laid  and  jointed  earthenware  glazed 
pipe.  In  order  to  prevent  deposits  the  mean  velocity  of  flow 
in  a  sewer  should  never  be  less  than  two  and  half  feet  per 
second  for  small  depths  of  flow.  In  order  to  ascertain  the 
mean  velocity  of  flow  in  such  sewer  pipe  when  flowing  only 
part  full,  the  coefficient  maybe  reduced  to  terms  of  hydraul 
icmean  depth  r,  in  feet  by  multiplying  m  in  terms  of  d  in 
feet  by  0.125,  or  by  dividing  by  8.  Then  in  terms  of  r  in  feet 

0004 
m  =!-g — —.0000."),  and  C— 141.42.     The  mean  velocity  of  flow 

in  a  circular  conduit,  or  in  a  pipe,  will  be   the    same  for  just 
half  full  as    for    full,    because-p-  is  the  same  for  half  full  as 

for  full. 

56— Table  of  Elementary  Dimensions  of  Pipes. 

TABLE  No.  25. 


Diam. 
In. 

Diam. 
Feet 

Area 
Sq. 
Feet 

U.S.  Gal. 
In  one 
Ft.Lgth 

Diam. 
In. 

Diam. 
Feet 

Area 
Sq. 
Feet 

U.S.  Gal 
In  one 
Ft.  Lsth 

K 

.0208 

.0003 

.0025 

4.V4 

3750 

.1104 

.8263 

% 

.0313 

.0008 

.0057 

4.% 

.3958 

.1231 

.9206 

y* 

.0417 

.0014 

.0102 

5 

.4167 

.1364 

1.020 

% 

.0521 

.0021 

.0159 

6 

.5 

.1963 

1.469 

X 

.0625 

.0031 

.0230 

8 

.6667 

.3491 

2.611 

% 

.0129 

.0042 

.0312 

10 

.8333 

.5454 

4.080    • 

i. 

.0833 

.0055 

.0408 

12 

.7854 

5.875 

i-k 

.1042 

.0085 

.0638 

14 

!l67 

1.069 

7.997 

l.V> 

.125 

.0123 

.0918 

16 

.333 

1.396 

10.440 

1.3£ 

.1458 

.0167 

.1249 

18 

.5 

1.767 

13.220 

2. 

.1667 

.0218 

.1632 

20 

.667 

2.182 

16.320 

2.M 

'1875 

.0276 

.2066 

22 

.833 

2.640 

19.75 

2-H 

.2083 

.0341 

.2550 

24 

2. 

3.142 

23.50 

2.X 

.2292 

.0412 

.3085 

26 

2.167 

3.6b7 

27.58 

3. 

.25 

.0*91 

.3672 

27 

2.25 

3.976 

29.74 

3.!4 

.2708 

.0576 

.43C9 

28 

2.333 

4.276 

31.99 

3  K 

.2917 

.0668 

.4998 

30 

2.5 

4.909 

36.72 

3.% 

.3125 

.0767 

.5738 

32 

2.667 

5.585 

41.78 

4. 

.3333 

.0873 

.6528 

34 

2.833 

6.305 

47.15 

4.M 

.3542 

498B 

.7369 

36 

3. 

7.069 

52.88 

231  cubic  inchee=d  U.  S.  gallon.    7.48052  U.  S.  gallons= 
1  cubic  foot. 


SULLIVAN'S  NEW  HYDRAULICS. 


163 


The  area  in  square  feet  of  a  pipe  is  the  same  as  the  contents 
Df  one  foot  in  length  of  the  pipe  in  cubic  feet.  Hence  by  an 
inspection  of  table  No.  25,  the  diameter  and  also  the  velocity 
required  to  carry  a  given  number  of  cubic  feet  or  of  U.  S.  gal- 
lons may  be  determined  at  once.  If  the  velocity  is  one  foot 
per  second  in  any  diameter,  the  discharge  in  cubic  feet  per 
second  will  equal  the  area  in  square  feet  of  that  diameter,  or 
the  discharge  in  gallons  per  second  will  equal  the  number  of 
gallons  in  one  foot  length  of  pipe.  For  a  discharge  of  2,  3,  4. 
etc  times  that  quantity,  the  velocity  must  be  2,  3,  4,  etc  feet 
per  second.  Tables  No.  16  and  17  and  19  will  show  the  slope 
or  head  required  to  generate  the  required  velocity,  and  also 
the  amount  of  head  that  will  be  neutralized  by  friction  for 
that  velocity.  The  dimensions  of  the  very  small  pipes  given 
in  table  No.  25  will  be  "found  convenient  in  designing  hy- 
draulic giants  and  nozzles,  and  in  selecting  small  service  pipes, 
and  discharge  pipes  for  small  pumpd.  See  also  Tables  22  and 
23,  and  26  and  27.  Square  inches  multiplied  by  .00695= 
square  feet. 

57.— Length  in  Feet  of  Small  Pipes  Required  to  Hold 
one  U.S.  Gallon  of  231  Cubic  Inches,  and  Areas  Given  in 
Square  Inches. 

TABLE  No.  26. 

(1  square  inch=.0069444  square  feet). 


Diam. 
In. 

Area 
Sq. 
Inches. 

Length  In 
Feet  to  hold 
1  Gallon. 

Diam. 
In. 

Area 
Sq. 
Inches. 

Length  In 
Feet  to  hold 
1  Gallon. 

U 

.0490875 

407.  43567  ;~0 

2.% 

5.412 

3.5570 

l/« 

.1963500 

98.0S92083 

2.% 

5.940 

3.2540 

2£ 

.4417875 

43.5729833 

2.X 

6.492 

2.9651 

1. 

.7854 

24.5098000 

3. 

7.069 

2.7225 

1  ii 

1.2271875 

15.6862001) 

3.^ 

7.670 

2.5097 

1  H 

1.7671500 

10.893iOOO 

3.ii 

8.296 

2.3200 

1  K 

2.4050 

8.0)41650 

3.% 

8.946 

2.1517 

l.X 

2.7610 

6.9721000 

3.V* 

9.fi21 

2. 

2. 

3.1416 

6.1274100 

3.X 

10.320 

1.8652 

2.H 

3.5470 

5.4270000 

8.V 

11.040 

1.7360 

2.?4 

3.9760 

4.8415000 

3.% 

11.790 

1.6166 

2  K 

4.9090 

3.9214000 

4. 

12.570 

1.5310 

Length  in  feet  to  hold  one  gallon  equals  velocity  in  feet 
per  second    required  to  discharge  one  gallon  per  second.    The 


164 


SULLIVAN'S  NEW  HYDRAULICS. 


velocity  must  be  7.5  times  as  great  to    discharge  one  cubic 
foot  per  second. 

A  4  inch  cast  iron  pipe  cannot  supply  one  fire  hydrant 
with  the  ordinary  supply  of  255  gallons  per  minute  without  a 
loss  of  head  in  such  pipe  of  nearly  one  foot  in  each  10  feet 
length  of  4  inch  pipe.  Add  to  this  the  friction  loss  in  the 
hydrant,  the  hose  and  the  nozzle,  and  the  resistance  of  the 
atmosphere  and  wind,  and  it  is  apparent  that  a  hydrant  will 
be  of  little  service  when  attached  to  a  four  inch  pipe  of  any 
considerable  length. 

58.— Decimal  Equivalents  to  Fractional  Parts  of  one 
Lineal  Inch. 

TABLE  No.  27. 


1-32=.  0312 1 
1-16=.  06250 
3-32=.  09375 
1-8  =.125 
1-8 -I- 1-32=.  15625 
1-8+1-16=. 1875 


18 +3-32=.  21875 
1-4=  .25 
1-4+1-32=. 28125 
1-4  +  1-16=. 3125 
1-4  +3-32=.  34375 
38=  .375 


|3  8+1-32=. 40625 
3  8+1-16=. 4375 
1 3-8 +3-32=.  46875 
1-2=  .5 
12+1-32=.  53125 
1-2 +  1-16=.  5625 


5-8  =.625 
5-8+1-16=. 6875 
3-4=.  75 
3-4 +  1-16=.  8125 
7-8=  .875 
7-8+3-32=. 96875 


Fractional  inches  in  equivalent  decimals  of  a  foot. 


Frac. 

Deci. 

Equiv 

Frac. 

Deci. 

Equiv 

Frac. 

Deci, 

Equiv 

Inch 

Inch 

dec  ft. 

Inch 

Inch 

dec  ft. 

Inch 

Inch 

dec  ft. 

1-32 

.03125 

.00?«04 

3-8 

.375 

.03125 

23-32 

.71875 

.059895 

1-16 

.0625 

.005208 

1332 

.40675 

.033854 

3-4 

.75 

0625 

3-32 

.09375 

.007X12 

7-16 

.4375 

.036458 

25-32 

.78125 

.065104 

1-8 

.125 

.010416 

15-32 

.46875 

.039062 

1316 

.8125 

.067708 

5-32 

.15625 

.010420 

1-2 

.5 

.041666 

27-32 

.84375 

.070312 

3-16 

.1875 

.015625 

17-32 

.53125 

.044i7 

7-8 

.875 

.072916 

732 

.21875 

.018229 

916 

.5625 

.046875 

2932 

.90625 

.07552 

14 

.25 

020833 

19-32 

.59375 

.049479 

15-16 

.9375 

.078125 

932 

.28125 

.023437 

5-8 

.62> 

.0-2083 

31-32 

.96875 

.080729 

5-16 

.3125 

.026041 

21-32 

.65625 

.054607 

1.00 

1.00 

.083333 

11-32 

.34375 

.028645 

11-16 

.6875 

.057291 

Tenths  of  one  foot  in  equivalent  inches. 


Foot 

Inches 

Foot 

Inches  1 

Foot 

Inches 

0.10 
0.20 
0.30 
0.40 

1  3  16 
2.3-8 
3.1932 
4.25-32      i 

0.50 
0.60 
0.70 
0.80 

6.00 
7.3-16     i 
8.3-8      ! 
9.19-32  ! 

0.90 

u 

10.25-32 
12.00 

SULLIVAN'S  NEW  HYDRAULICS. 


165 


59— Tables  for  Converting  Measures, 

TABLE  No.  28.    Lineal  Measure. 


Inch's 

Feet 

Yards 

Fath. 

Rods 

Miles 

Metres 

l 

12 
36 
72 

198 
7920 
63360 

.083333 
1 
3 
6 
16l/s 
660. 
52SO. 

.02778 
.33333 
1. 
2. 

220." 
1760. 

.013889 
.16666 
.5 
1. 

no!  4 

880. 

.005051 
.060606 
.181818 
.363636 
1. 
40. 
320. 

.000016 
.000189 
.000563 
.001136 
.003125 
.125 
1.0 

.0254 
.304797 
.914392 
1.82878 
5.02915 
201.166 
1609.33 

TABLE  No.  29.    Land  Measure  (Lineal). 


Inch's 

Links 

Feet 

Yards 

Chains 

Miles 

Metres 

7  23-25 
12 
36 
792 
63360 

.1261261 
1. 
1  17-33 
4  6-11 

100. 
8000. 

.083333 
.066666 
1. 
3. 
66. 
5280. 

.0277778 
.222222 
.333333 
1 
22. 
1760. 

.0012626 
.01 
.0151515 
.0454545 
1. 
80. 

.0000158 
.0001£5 
.00"1894 
i  01X15682 
.0125 
1. 

.0254 
.201166 
.304797 
.914392 
20.1166 
1609.33 

TABLE  No.  30.*    Metrical  Equivalents.     Lineal  Measure. 


Inches 

Feet 

Yards 

Rods 

Chains 

Miles 

1  Millimeter= 
1  Centimeter= 
1  Meter 
1    Kilometer= 

.03937 
.393704 
39.370432 

.003281 
.0328(19 
3.2*0869 
3280.8693 

.001094 
.Olf'936 
1.093623 
1093.6231 

.001988 
.198841 
198.84057 

.04971 
49.710141 

.000621 
.621377 

Milli- 
meters 

Centi- 
meters 

Meters 

Kilo- 
meters 

Inch    = 
Foot    = 
Yard    = 
Rod     = 
Chain  = 
Mile    = 

25.399772 
304.79727 
914.391795 
5029.15487 

2.539977 
30.47973 
91.43918 
502.9ln49 
2011.66195 

.253998 
.304797 
.914392 
5.029155 
20.11R62 
1609.32956 

.0003048 
.0009144 
.00"02915 
.02011662 
1.60933 

ine. 


*See  "Rules  and  Tables"  page  92,  by  Prof.  W.  J.  M.  Rank- 


SULLIVAN'S  NEW  HYDRAULICS. 


167 


TABLE  No.  33.     Cubic  Measure. 


Cubic  Inches 


1.0 

1728.0 
46656.0 


Cubic  Feet     I  Cubic   Yards  ICubic  Meters 


.0005788 
1.0 
27. 


.00000214 
.037037 
1.0 


.000016387 

.0283161 

.764534 


231  cubic  inches=l  U.S.  gallon.  7.48052  U.  S.  gallons^ 
1  cubic  foot. 

The  actual  weight  of  1  U.  S.  gallon  of  water  at  its  maxi- 
mum density  is  8.345008  pounds.  The  weight  is,  however, 
adopted  by  law  as  8.33888  pounds  avoirdupois. 

1  U.  S.  gallon=.13368  cubic  foot.  1  cubic  foot  per  second 
=448.8312  gallons  per  minute,  or  26929.872  gallons  per  hour, 
or  646316.928  gallons  per  24  hours.  1  cubic  foot  per  second= 
60  cubic  feet  per  minute,  or  3600  cubic  feet  per  hour,  or  86400 
cubic  feet  per  24  hours.  This  will  cover  one  acre  of  ground 
to  a  depth  of  1  98347  feet,  or  1.98347  acres  to  a  depth  of  one 
foot  iu  2i  Lours,  or  supply  200  gallons  per  person  per  24  hours 
for  3,231 58  persons.  An  8  inch  pipe  will  carry  it  at  a  velocity 
of  2.864  feet  per  second. 

TABLE  No.  34.     Metrical  Equivalents. — Cubic  Measure. 


Cubic  In. ;U.  S.  Gal-|CubicFt,'Cubic  yd|Perches: 


leu.  centimtr. 
1  cu.  rlecin.etr. 
1  cubic  meter 

.061025386  !   .000264179      .000035316 
61.025386    ,   .264179           .035316 
6  '025.  386     [  264.179        [  35.316 

.001307986  |  .001426893 
1.307986      |  1.426893 

Cubic  Cent. 

Cubic  Deem. 

Cubic    Meters 

1  cubic  inch 
1  U.  8.  gallon 
1  cubic  foot 
1  rubic  yard 
1  Perch 

16.386623    . 
3785.31 
28316.0844 

.016386623 
3.78531 
28.3160844 
764.5343 
700.82309 

.00378531 
.0283160844 
.7«J5343 
.70082309 

1  Perch=24.75  cubic  feet. 

TABLE  No.  35.— Pressure.     (Thurston). 


Pounds  per  sq. 
Inch. 

Kilograms  per  [Kilograms  per 
Sq.  CentimeterjSq.  Centimeter 

Pounds  per. 
sq.  Inch. 

1.0 

.07030S27           1               1.0 

14.22308 

REMARK.— The  foregoing  conversion  tables  are  given  in 
order  that  the  formulas  may  be  used  and  coefficients  deter- 
mined either  in  English  or  metrical  terms. 


SULLIVAN'S  NEW  HYDRAULICS. 
TABLE  No.  31.— Square   Measure. 


Sq. 
Inches 

Square 
Feet 

Square 
Yards 

Square 
Rods 

Square 
Roods 

Square 
Acres 

Square 
Metres 

i. 

144. 

1296. 
39204. 
1568160. 
C272640. 

.00(59444 
1. 
9. 
272.  k 
lftS90. 
435(10 

.0007716 
.1111111 
1. 

30.  yt 

1210. 
4840. 

.0000255 
.0036731 
.0330579 
1. 
40. 
160. 

.00000064 
.  .OIX;0918 
.0008264 
.025 
1. 
4. 

.00000016 
.00002:$ 
.0002050 
.00620 
.25 
1. 

.0006452 
.0929013 
.836112 
25.292 
1011.6P6 
4046.782 

AcresX-C015625  = square  miles.  1  square  mile=27,878,400 
square  feet,  or  3C97600  square  yardb,  or  640  acres,  or  one  sec- 
tion. One  acre=10  square  chains.  Tho  length  of  one  chain 
is  66  feet,  or  four  rods.  This  Gunters  chain  has  fallen  into 
disuse,  and  a  steel  tape  100  feet  length  is  used  instead.  Areas 
are  taken  in  square  feet,  and  when  divided  by  43,560,  are  re- 
duced to  acres. 

TABLE  No.  32*— Metrical  Equivalents.     Square  Measure. 


Square 
Inches 

Square 
Feet 

Square 
Yards 

Acres 

Square 
Miles 

1  sq.  Centime- 

ter= 

.165003 

.00107641 

.00012 

1   eq.  Decime- 

ter= 

15.500309 

.107541 

.011960115 

1    sq.  Meter= 

1550.030916 

10.7641 

1.1960115 

.00024711 

1  eq.  Dekame- 

ter=l  Are= 

15r-003.0916 

1076.41 

119.  C011  5 

.02411 

.00003861 

1  sq.  Hectome- 

ter=l  Hectare 

1    Kilometer= 

1076.41 

11960.115           2.4711 
1196011.5       247.11 

.003861 
.3861 

Sq.  Centi- 
meters 

Sq.  Deci- 
meters 

Sq.  Meters. 

Sq.  Dekame- 
ters  or 
Ares 

ISq 
Inch= 
1  Square 
Foot= 
1  Square 
Yard= 

6.451484 
929.013728 
8361.123554 

.06451484 
9.29013728 
83.61123554 

.0006451484 
.0929013728 
.8361123554 

.000929013728 
.008361123554 

'See  "Conversion  Tables,"  page  40,  by  Prof.  Thureton, 
and  Trautwine's  "Civil  Engineer's  Pocket  Book",  page  78, 
Rankine's  "Rules  and  Tables,"  p.  p.  110-114. 


CHAPTER  V 


Of  Water  Powers,  Power  Mains  and  Pipe   Lines. 

Work  is  expressed  in  units  of  weight  lifted  through  one 
unit  of  height;  as  in  pounds  lifted  one  foot,  called  foot  pounds. 
Here  there  is  no  reference  to  the  units  of  time  consumed 
Power  is  expressed  in  units  of  work  done  in  one  unit  of  time; 
as  in  pounds  lifted  one  foot  in  one  second  of  time,  called  foot 
pounds  per  second. 

One  horse  power  is  a  conventional  quantity  equal  to  550 
foot  pounds  per  second,  or  to  550  pounds  lifted  one  foot  in 
one  second,  or  to  one  pound  lifted  550  feet  per  second. 

As  there  are  60  seconds  in  one  minute  of  time,  theexpres 
ion  of  horse  power  in  terms  of  foot  pounds  per  minute  would 
be  550X60=33,000.  or  in  foot  pounds  per  hour  it  woud  be  33,- 
000X60=1,980,000. 

One  pound  of  water  falling  one  foot  does  work  equal  to 
that  of  raising  one  pound  one  foot  high.  Hence  the  number 
of  pounds  of  water  falling  in  one  second  multiplied  by  the 
distance  fallen  in  feet  will  equal  the  number  of  foot  pounds 
per  second,  and  as  550  foot  pounds  per  second  equal  one  horse 
power,  the  totil  number  of  foot  pounds  per  second  divided  by 
550  will  equal  the  horse  power  of  the  water.  Expressed  as  a 
formula,  we  have 

cubic  f  eet  per  Bec.X  weight  of  one  cubic  footX  Head  or  fall  in  feet. 
H'P-=-  550 

The  weight  of  one  cubic  foot  of  water  at  its  maximum 
density  is  62.5  Ibs.  This  is  the  weight  always  assigned,  in  or- 
dinary cases,  to  one  cubic  foot  of  water.  The  formula  may 
therefore  be  written 

62  5 
H.  P.=  --Xcubic  feet  per  second  X  head  or  fall  in  feet. 


62  5 
If  we  take   the  quotient  of  --  =.3136363,  we  have, 


SULLIVAN'S  NEW  HYDRAULICS.  169 

-  H.  P.=.1136363Xhead  in  feetXcubic  feet  per  second..  .97 

61.— Formula  for  Cubic  Feet  Per  Second  Required  to 
Generate  a  Given  Horse  Power. 

When  the  net  head  or  fall  in  feet  is  given,  then  the  cubic 
feet  per  second  required  to  develop  any  required  horse  power 
will  be 

Horse  Power  Desired 

Cubicfeet  per  sec.= jll36ae3xHeBd  in  Feet <98> 

62.— Formula  for  Net  Head  or  Fall  in  Feet  Required 
to  Develop  a  Given  Net  Horse  Power. 

The  efficiency  of  a  water  wheel  or  other  machine  is  the 
ratio  of  effective  power  recovered  from  it  to  the  total  power 
applied  to  it.  To  find  the  efficiency,  divide  the  effective  pow- 
er delivered  by  the  machine,  by  the  total  power  applied  to  it. 
The  quotient  is  the  efficiency.  If  a  water  fall  of  100  horse 
power  is  applied  to  a  turbine  and  the  turbine  develops  80 

horse  power,  then  the  efficiency  of  the  turbine  is  E= j^  =  80 
per  cent. 

The  efficiency  of  the  motor  being  given,  then  the  net  head 
or  fall  in  feet  required  to  develop  a  given  net  horse  power, 
will  be 

Desired  net   H.  P.-nper   cent  efficiency  of  motor 
.1136363XCubic  Feet  per  Second, 

(99) 

63.— Head  of  Water  Defined.—  By  the  term  head  is 
meant  the  difference  of  level  between  the  surface  of  the  water 
in  the  reservoir  or  head  race,  and  the  water  surface  in  the  tail 
race,  to  which  must  be  added  the  head  due  to  the  mean  vel- 
ocity of  flow  in  the  head  race  or  stream  above  the  fall. 

The  head  due  to  the  velocity  in  the  head  race  is  H=— — 

(See  formula  (7)  Chap.  I.)  In  a  pipe  or  power  main  the  total 
head  is  equal  to  the  difference  of  level  between  the  water  sur- 
face at  the  intake  end  of  the  pipe  and  the  upper  surface  of 
the  jet  at  discharge,  and  the  net  head  is  equal  to  the  total 


170  SULLIVAN'S  NEW  HYDRAULICS. 

head  lees  the  amount  of  head  neutralized  by  friction  or  re- 
sistance in  the  pipe.  Hence  in  a  pipe  or  power  main  the  loss 
of  head  by  friction  must  be  first  deducted  from  the  total 
head  in  order  to  ascertain  the  effective  head  at  discharge. 

64.  —  To  Find  the  Diameter  in  Feet  of  Pipe  Required 
to  Carry  a  Given  Quantity  of  Water  with  a  Given  loss 
of  Head  in  Feet, 

Where  the  total  head  is  known,  and  it  is  desired  to  lay  a 
pipe  of  such  diameter  as  will  convey  a  given  number  of  cubic 
feet  per  second  with  a  predetermined  loss  of  head  by  friction, 
so  that  a  given  net  head  will  be  secured  at  discharge,  such  di- 
ameter in  feet  may  be  found  as  follows: 

x'  .......................  (32) 


For  ordinary  clean  cast  iron  pipe  n=.0003938  in  terms  of 
diameter  in  feet. 

Hence  the  formula  reduces  to 


and, 

'..  whence. 


h" 

4X<1S (100) 


h" 

If  it  is  an  asphaltum  coated  pipe,  then  take  n=00032  in 
terms  of  diameter  in  feet,  and  the  formula  for  finding  the  di- 
ameter required  to  carry  a  given  quantity  in  cubic  feet  per 
second  with  a  given  total  loss  of  head  in  feet  by  friction  in 
the  entire  length  of  pipe  line,  will  be 
11 

V 

In  these  formulas 

d=diameter  of  required  pipe  in  feet 
q=cubic  feet  per  second  it  is  to  discharge 
Z=total  length  of  pipe  in  feet 
h"=head  in  feet  lost  by  friction  in  total  length,  I, 


SULLIVAN'S  NEW  HYDRAULICS.  171 

EXAMPLE   OF  THE   USE   OF   THESE    FORMULAS. 

There  is  a  total  fall  of  100  feet  in  a  distance  of  3,000  faet. 
A  diameter  of  asphaltum  coated  pipe  is  desired,  which  will 
convey  one  cubic  foot  of  water  per  second  with  a  loss  of  head 
not  exceeding  6  feet,  so  that  there  shall  remain  an  effective 
head  of  94  feet,  at  discharge  while  one  cubic  foot  per  second 
is  being  drawn  from  the  pipe  at  its  lower  end.  By  the  above 
formula 


d=ll/  .000000269X  J  X9000000  _  11 
V  36  V 

feet, 

This  diameter  has  an  area  =d2X-7p54=.48078  square  feet. 
The  mean  velocity  required  to  discharge  one  cubic  foot  per 

second  in  this  diameter  would  be,  v= ~ 48078  ~  2.08  feet 

per  second. 

The  resu't  may  therefore  be  tested  by    the   formula  for 
loss  of  head, 


And  we  have, 

.00032 
h"=     692   X3000X4.3264-6.00  feet  head  lost. 

It  will  be  understood  that  the  area  at  discharge  is  such 
that  it  will  admit  of  no  greater  discharge  under  the  given  net 
head  than  the  quantity  q.  The  manner  of  discharge  may  be 
through  other  small  pipes  tapped  into  the  main  if  it  is  a 
water  works  system,  or  the  discharge  may  be  through  a  re- 
ducer or  nozzle  if  the  pipe  is  used  as  a  power  main  for  driv- 
ing water  wheels,  or  the  discharge  maybe  full  and  the  total 
head  lost  except  the  velocity  head,  as  may  be  desired. 

65.  —  To  Find  the  Area  and   Diameter   of  the  Nozzle 
Tip  or  Aperture  Required  to  Discharge  a  Given    Quantity. 

If  there  is  a  simple  tip  on  the  end   of   the  pipe  made  in 
the  form  of  the  contracted  vein  which  reduces  the  diameter 


172  SULLIVAN'S  NEW  HYDRAULICS. 

at  discharge,  there  will  be  a  very  small  loss  of  head  by 
friction  in  efflux  from  the  tip.  The  area  in  square  feet  of  the 
required  aperture  in  such  tip  will  be  found  as  follows:  As- 
sume the  diameter  of  the  pipe  to  be  .7824  feet,  and  net  head 
at  discharge  to  be  94  feet,  as  in  the  preceding  section,  and  the 
quantity  to  be  disharged  as  one  cubic  foot  per  second. 

The  velocity  that  will  be  generated  by    this    net  head  at 
discharge  will  be 

v  =  1/iTgTl=8.0251/94~~=77.8052  feet  per  second. 

Now,  q=  areaX velocity  ^ay^g  H.    Whence  a  = 

q  10 

1/2    H=  77  8052~='Q128526  8<luare  feet-  The  diameter  in  feet 

is  then 


='128  foot=1«636  inchee  diam' 


=  .7854 

eter. 

See  table  No.  27,  §  58. 

If  the  discharge  is  to  be  through  a  nozzle  or  reducer  of 
several  feet  length,  there  will  be  considerable  loss  of  head  by 
friction  in  such  nozzle  or  reducer,  for  which  allowance  must 
be  made.  This  loss  will  depend  upon  the  length  of  the  con- 
vergent reducer  or  nozzle  and  its  mean  or  average  diameter  as 
well  as  its  smoothness  of  internal  circumference,  and  the 
square  of  the  velocity  through  it.  We  have  seen  heretofore 
(§37,  39)  that  the  loss  by  friction  in  a  convergent  or  conical 
pipe  is  nine  times  as  great  as  the  loss  in  a  pipe  of  uniform 
diameter  equal  to  the  mean  diameter  of  the  convergent  pipe. 
It  is  therefore  evident  that  such  convergent  pipes,  reducers 
or  nozzles  should  be  as  short  as  possible.provided  they  do  not 
converge  more  rapidly  than  one  inch  in  a  length  2.33  inches, 
which  would  make  them  conform  to  the  form  of  the  con- 
tracted vein.  Assuming  the  diameter  of  the  base  of  the  noz- 
zle to  be  the  same  as  the  diameter  of  the  pipe  it  is  to  join,  and 
that  the  net  head  at  the  base  of  the  nozzle  is  94  feet,  and 
that  the  reducer  or  nozzle  is  to  be  6  feet  in  length,  and  is  re- 
quired to  discharge  one  cubic  foot  per  second  under  this  net 


SULLIVAN'S  NEW  HYDRAULICS.  173 

head  at  the  base,  the  problem  now  is  to  determine  the  area 
and  diameter  of  the  small,  or  discharge  end  of  the  nozzle  so 
that  it  shall  discharge  this  given  quantity  per  second  under 
the  given  head  at  its  base. 

This  will  require  one  or  more  approximations,  for  the 
reason  that  as  the  mean  diameter  of  the  proposed  nozzle  is 
yet  unknown  we  have  no  means  of  knowing  the  friction  loss 
that  will  occur  in  the  nozzle,  and  hence  do  not  kuow  the  value 
of  the  net  effective  head  at  the  point  of  final  discharge  from 
the  nozzle. 

For  first  approximation  assume  that  there  will  be  three 
feet  head  lost  in  the  nozzle,  leaving  an  assumed  effective 
head  at  discharge  of  94—3=91  feet.  The  velocity  of  dis- 
charge under  the  net  head  of  91  feet  will  be  ^8.025/91= 
76.5535  feet  per  second.  Then  the  area  in  square  feet  re- 
quired to  discharge  the  quantity  q,  in  cubic  feet  per  second, 

will  be  a=^!=g=:  7^35  =  .01306276  square  feet.  The  di- 
ameter in  feet  answering  to  this  area  in  square  feet  is 


eter  in  inches  is  therefore   .129X12=1  548  inches.    (See  §58. 
Table  27). 

For  first  approximation  we  have  then  the  following  di- 
mensions of  the  nozzle:  —  Greatest,  or  butt  diameterr=.7824 
foot  =9.388  inches.  Smallest,  or  discharge  diameter  =.129 
foot=l  548  inches.  Total  length  of  nozzle—  6  feet.  The 
average  or  mean  diameter  of  the  nozzle  is  therefore, 

Mean  d='    "  "*"'  -  =  .4557  foot,  or  5.4684  inches. 


Now,  two  tests  must  be  applied  to  this  nozzle  in  order  to 
ascertain  whether  or  not  it  will  fulfill  the  required 
conditions:— 

(1)  It  must  be  tested  by  the  formula  (§39)  for  friction  lose 
in  nozzles  in  order  to  ascertain  the  actual  loss  of  head  that 
will  occur  while  discharging  the  given  quantity. 

(2)    It  must  then  be  tested  to  ascertain   whether  or  not 


174  SULLIVAN'S  NEW  HYDRAULICS. 

it  will  discharge  the  given  quantity  under  the  conditions 
actually  existing.  If  it  fails  to  meet  the  requirements,  fur- 
ther approximation  must  be  made. 

(1)  TEST  FOR  LOSS  BY  FRICTION. 

The  formula  for  loss  of  head  in  feet  by  friction  in  conver- 
gent pipes  and  nozzles  is 

h"=     °    X/X9V.     (See  §§37,39). 

In  which 

h"=head  in  teet  lost  by  friction 
i=rlength  in  feet  of  convergent  pipe  or  nozzle* 
d—  average  or  mean  diameter  of  nozzle 
v=mean  velocity  in  the  mean  diameter  of  the  nozzle 
In  the  nozzle  we  are  now  considering  the  mean   diameter 
is   .4557  foot,  and    the    nozzle  is    required   to  discharge  one 
cubic  foot  per  second.    Hence  the  required  mean   velocity  in 
feet  per  second  through  this  mean  diameter  to  cause  the   dis- 
charge of  one  cubic  foot  per  second  will  be 

v=-9_=  --  —  -  =  _  L°_=6.1312  feet  per  second. 
a        (.4557)sX-7854      .1631 

Assuming  the  nozzle  to  be  made  of  very  dense,  solid 
smooth  cast  iron,  the  friction  coefficient  will  be  n  —  .0003623  in 
terms  of  diameter  in  feet.  Applying  the  above  formula  for  loss 
by  friction  in  this  nozzle  while  discharging  one  cubic  foot  per 
second,  the  velocity  in  the  mean  diameter  being  6.1312  feet 
per  second,  and  we  find  the  actual  loss  of  head  in  the  nozzle 
to  be 


Hence  at  the  point  of  discharge  the  effective  head  would 
be  94—2.40=91.60  feet,  whereas  we  had  assumed  that  it 
would  be  probably  91.00  feet.  But  as  the  assumed  loss  of  head 
(3  feet)  and  the  actual  loss  (2.40  feet)  are  BO  nearly  equal,  we 
will  now  apply  the  test  for  quantity  discharged  under  the 
actual  conditions.  For  this  purpose  we  have  the  following: 

Area  of  smallest  diameter  at  discharge=.01306276  square 
feet. 


SULLIVAN'S  NEW  HYDRAULICS.  175 

Effective  head  at  point  of  discharge  from  nozzle— 91.60 
feet. 

Velocity  due  to  this  net  head,  v=!/2gH.  =  8.025v/91.bp= 
76.807275  feet  per  second.  Quantity  discharged  q,  will  be 

q=a  v^.01306276=76.807275=:1.003315  cubic  feet  per  sec- 
ond. 

If  a  closer  result  is  desired,  the  smallest  diameter  may  be 
reduced  by  1-16  inch  and  all  the  foregoing  tests  be  again  ap- 
plied to  the  new  proportions  of  the  nozzle  thus  changed. 
Table  No.  21,  §  40,  will  be  of  assistance  in  such  calculations. 

66. — Pipe  Lines  of  Irregular  Diameter. — Where  the  head 
or  pressure  is  due  to  the  slope  or  inclination  of  a  pipe  line, 
and  not  to  a  pump,  there  will  be  very  little  pressure  within 
the  pipe  in  the  upper  portion  of  the  line.  In  such  cases 
large  diameters  with  thin  shells  may  be  adopted  in  the 
upper  part  of  the  line  where  the  pressure  is  small.  As  the  line 
proceeds  down  the  slope  and  the  pressure  increases,  the  di- 
ameter is  diminished  and  the  pipe  shell  increased  in  thickness 
in  proportion  to  the  increase  of  pressure. 

If  a  pipe  line  is  of  uniform  diameter  and  is  laid  on  a  uni- 
form grade  and  has  a  full  and  free  discharge,  there  will  be  no 
radial  pressure  in  the  pipe  at  any  point  except  the 
very  small  pressure  due  to  the  vertical  depth  of  the  diameter. 
In  this  cas?  there  is  no  object  in  increasing  the  thickness  of 
pipe  shell  at  its  lower  end,  because  the  total  head  or  pressure, 
under  these  conditions,  will  be  converted  into  velocity  of  flow, 
with  the  exception  of  the  amount  of  head  lost  by  friction,  and 
as  the  velocity  head  or  velocity  pressure  is  always  parallel 
to  the  pipe  walls,  it  does  not  tend  to  burst  the  pipe. 

Where  a  given  pressure  or  head  is  to  be  maintained  at 
the  lower  end  of  the  pipe,or  at  any  point  along  its  length, while 
a  given  supply  of  water  is  being  drawn  from  it  for  domestic 
purposes,  or  for  driving  water  wheels,  the  capacity  of  the 
pipe  must  be  such  that  the  mean  velocity  of  flow  in  it  while 
delivering  the  given  supply,  will  not  cause  a  loss  of  head  by 
friction  exceeding  a  predetermined  amount.  The  discharge 
permitted  from  such  pipe  must  therefore  bo  regulated  by 


176  SULLIVAN'S  NEW  HYDRAULICS. 

the  area  of  discharge  BO  that  it  will  not  exceed  the  givea 
quantity.  If  the  lower  end  of  a  pipe  line  be  entirely  closed 
so  there  can  be  no  discharge  from  it  and  no  velocity  within 
it.the  pressure  at  any  point  along  the  line  will  be  that  due  to 
the  total  head  up  to  that  point,  which  will  be  equal  to  the 
difference  in  level  between  the  given  point  in  the  pipe  and 
the  water  surface  in  the  reservoir  or  source  of  supply.  The 
pressure  at  the  lower  end  will  be  that  due  to  the  total  head 
in  the  pipe  line.  If  a  small  orifice  be  opened  in  the  lower  end 
of  the  pipe,  it  will  at  first  discharge  with  a  velocity  due  to  the 
total  head,  but  this  discharge  will  cause  a  small  velocity  to 
be  generated  throughout  the  length  of  the  entire  pipe,  and 
this  velocity  will  cause  a  small  friction  with  the  pipe  walls 
which  will  reduce  the  head  by  the  amount  of  the  friction 
thus  generated,  and  thus  slightly  check  the  velocity  of  dis- 
charge through  the  orifice.  The  smaller  the  orifice  relatively 
to  the  area  and  capacity  of  the  pipe,  the  smaller  will  be  the 
velocity  in  the  body  of  the  pipe  to  supply  the  quantity  being 
discharged;  and  as  the  loss  of  head  by  friction  is  as  the  square 
of  the  velocity,  the  smaller  the  velocity  becomes,  the  smaller 
the  loss  by  friction  will  become.  If  the  orifice  is  enlarged  so 
that  it  may  discharge  a  greater  quantity  per  second,  then 
the  velocity  in  the  body  of  the  pipe  must  increase  proportion- 
ately and  loss  of  head  or  pressure  will  also  increase  as  the 
square  of  this  greater  velocity.  If  the  entire  end  of  the  pipe 
be  opened  so  that  the  discharge  is  entirely  free,  then  the 
total  head  will  be  lost  in  friction  due  to  the  consequent  high 
velocity,  except  the  small  portion  of  the  total  head  which  re- 


mains  to  generate  the  velocity,  and  which  is  h  v=g^|.      It  is 

evident  then  that  if  head  or  pressure  is  to  be  preserved  the  di- 
ameter and  area  of  the  pipe  must  be  sufficient  to  convey  the  re- 
quired quantity  at  a  low  velocity,  and  the  pipe  must  not  be 
permitted  to  discharge  at  anything  like  its  full  capacity.  As 
loss  of  head  or  pressure  is  directly  as  the  roughness  of  the 
pipe,  and  directly  as  the  length,  and  inversely  as  j/d3,  it  is 
necessary  to  take  into  account  not  only  the  diameter  and  ve- 


SULLIVAN'S  NEW  HYDRAULICS.  177 

locity  but  also  the  length  of  pipe,  and  the  nature  of  the  pipe 
walls  with  regard  to  smoothness  or  roughness,  and  probable 
future  deterioration.  The  chemical  qualities  of  the  water 
which  is  to  flow  through  a  pipe,  and  the  effect  they  have  upon 
different  classes  of  pipe  and  pipe  coatings  should  be  care- 
fully ascertained  before  the  pipe  is  selected.  Some  waters, 
apparently  almost  pure,  will  corrode  a  pipe  in  a  very  short 
time  to  such  an  extent  as  to  reduce  its  capacity  by  nearly  one 
half.  A  pipe  line  made  up  of  different  diameters,  gradually 
decreasing  as  the  slope  increases,  designed  to  convey  a  given 
quantity  and  to  maintain  a  given  pressure,  is  some- 
times less  expensive  than  a  pipe  line  of  uniform 
diameter.  The  velocities  in  the  different  diametera 
of  such  irregular  pipe  lines  will  be  inversely  as  the 
areas  of  the  different  diameters  and  the  friction  loss  in 
each  section  will  be  as  the  square  of  the  velocity  in  that  sec- 
tion and  inversely  as  ;/d8.  The  loss  of  head  in  such  a  line 
must  be  calculated  separately  for  each  different  diameter. 
In  case  the  line  is  divided  into  divisions  of  equal  lengths,  and 
each  division  is  of  a  constant  diameter  but  of  a  different  di- 
ameter from  the  rest  of  the  line,  the  mean  diameter  of  the 
whole  line  cannot  be  adopted  for  such  calculations,  because 
the  mean  of  all  the  velocities  in  the  different  diameters  will 
greatly  exceed  the  mean  velocity  in  a  pipe  of  uniform  diameter 
equal  to  the  mean  diameter  of  the  line  composed  of  different 
diameters.  As  the  friction  is  as  the  square  of  the  velocity,  it 
is  evident  that  it  will  be  much  greater  in  the  line  of  decreas- 
ing diameters  than  in  a  pipe  of  uniform  diameter  equal  to  the 
mean  diameter  of  the  former.  Where  the  saving  of  head  or 
pressure  is  a  principal  object  there  are  only  a  few  cases  in 
which  it  is  cheaper  or  advisable  to  adopt  large  diameters  for 
the  upper  portion  of  the  line  and  smaller  ones  for  the  lower 
portion.  What  is  saved  in  the  cost  of  constructing  such  line 
is  lost  in  head  or  pressure,  which  maybe  of  more  value  than 
the  difference  in  cost  between  the  two  kinds  of  pipe  line.  For 
example  a  pipe  line  5,000  feet  in  length,  made  of  lap  welded 

Sipe   and  thoroughly  coated   with   asphaltum,  in  which  the 
ret  1,000  feet  length  has  a  diameter  of  three  feet,  the  second 


178  SULLIVAN'S  NEW  HYDRAULICS. 

1,000  feet  has  a  diameter  of  2.75  feet,  the  third  1,000  feet  has  a 
diameter  of  2.5  feet,  the  fourth  a  diameter  of  2  feet,  and  the 
fifth  a  diameter  of  1.5  feet,  while  discharging  8  cubic  feet  per 
second,  will  have  velocities  and  losses  of  head  in  the  different 
diameters  as  follows: 

In  section  No.  1,  ¥=_!.=:     ^—=1.1178  feet,  h"=     .072  feet 
a      7.069 

In  section  No.  2,  v=-*L=     *L_ =1.3470  feet,  h"=     .119" 
a       594 

In  section  No.  3,  v=-3_=-JL=1.6300  feet,  h"=     .202  " 
a,      4  909 

In  section  No.  4,  v=-5-=_ —  =2.5400 f eet,  h"=  .680  " 
a      3.1416 

In  section  No.  5,  v=^L=    8    =4.5200  feet,  h"=  3.350  " 
a      1767 


4.423 

The  loss  of  head  for  this  small  discharge  will  be  4.423  feet 
in  the  line  of  different  diameters,  and  the  mean  of  all  the 
velocities  in  the  different  diameters  will  be  2  2771  feet  per  sec- 
ond. 

Now  if  the  sum  of  these  five  different  diameters  is  divided 
by  5  we  have  the  mean  diameter  2.35  feet.  The  area 
of  this  mean  diameter=4.3374  square  feet.  Conse- 
quently if  the  entire  pipe  line  had  been  of  the  uniform  diam- 
eter of  2.35  feet,  the  necessary  velocity  through  it  to  cause  a 
discharge  of  8  cubic  feet  per  second  would  be 

1.84442  feet,  and  the  total  loss  of  head 


_ 
4.3374 

would  have  been  h"  =1.424  feet.  (For  this  class  of  pipe  n= 
.0003).  As  the  friction  is  inversely  as  i/d8,  and  also  directly 
as  v2,  it  is  apparent  that  a  small  increase  of  the  discharge 
would  greatly  increase  the  loss  by  friction  in  sections  No.  4 
and  5  of  the  irregular  diameter. 

67—  A  Power  Main  with   Nozzle,  and    Water  Wheel  to 
Run  at  a  Given  Speed  and  Develop  a  Given  Power.— 

In  mountainous  regions  are   many   small   torrents,   the 


SULLIVAN'S  NEW  HYDRAULICS.  179 

sources  of  which  are  at  such  great  altitudes  as  to  afford 
almost  any  head  desired  when  the  stream  is  confined  within 
a  pipe  or  power  main  so  as  to  preserve  the  head  or  pressure 
by  regulating  the  velocity  of  flow.  Where  the  quantity  of 
water  is  small  and  the  head  is  great,  an  impulse  and  re- 
action water  wheel  will  be  much  more  efficient  and  satisfac- 
tory than  a  turbine.  The  loss  of  head  in  a  power  main  de- 
pends upon  the  velocity  of  flow  through  it  and  upon  its 
length,  diameter  and  smoothness  and  freedom  from  bends. 
The  velocity  is  governed  by  the  quantity  of  wate'r  the  main  is 
permitted  to  discharge,  and  the  quantity  discharged  is  gov- 
erned by  the  area  of  discharge  at  the  point  of  the  nozzle  and 
by  the  effective  head  at  discharge. 

The  greater  the  length  of  the  pipe  line,  the  smaller  the 
velocity  must  be,  for  the  loss  of  head  by  friction  is  directly  as 
the  length  and  as  the  square  of  the  velocity.  Such  power 
mains  or  pipelines  are  usually  constructed  of  riveted  pipe 
made  of  steel  or  wrought  iron  plate.  The  pipe  is  made  in  any 
convenient  lengths  for  transportation,  or  is  made  on  the 
ground  where  it  is  to  be  laid.  After  it  is  riveted  into  lengths 
it  is  thoroughly  coated  by  being  submerged  in  a  tank  of  hot 
coating  compound  composed  of  80  per  cent  asphaltum  and  20 
per  cent  crude  petroleum  which  is  maintained  at  a  tempera- 
ture of  about  300  degrees  Fahr.  The  pipe  is  allowed  to  re- 
main submerged  in  the  hot  bath  until  the  pipe  metal  attains 
the  same  temperature  as  the  bath.  It  is  then  withdrawn 
from  the  bath  and  allowed  to  cool.  In  some  cases  coal  tar 
45  per  cent  and  asphaltum  55  per  cent  is  used  as  a  coating 
with  fair  results. 

The  quality  or  purity  of  the  asphaltum  used  will  deter- 
mine the  best  proportion  of  asphaitum  to  crude  petroleum  to 
use  in  the  compound.  The  per  cent  of  petroleum  required 
varies  from  15  to  20.  After  the  compound  has  been  heated 
and  thoroughly  mixed  and  incorporated  it  should  be  tested 
by  dipping  into  it  a  small  sheet  of  the  pipe  metal  and  allow- 
ing it  to  remain  for  ten  minutes  in  the  hot  bath.  It  is  then 
withdrawn  and  placed  in  a  large  vessel  of  cold  water  and  al- 


180  SULLIVAN'S  NEW  HYDRAULICS. 

lowed  to  cool.  If  the  coating  is  too  soft  after  cooling  and  has 
a  tendency  to  run  or  wrinkle,  there  is  too  much  oil  in  it,  and 
the  quantity  of  asp  laltuin  should  be  increased.  If  it  is  a  mix- 
ture of  coal  tar  and  asphaltum  the  coating  will  be  too  brittle 
and  easily  knocked  off  with  a  hammer  if  the  proportion  of  tar 
is  too  great  to  that  of  asphaltum. 

In  any  case  the  coating  should  be  tough  and  elastic  and 
should  adhere  to  the  metal  similar  to  paint.  If  the  bath  is 
too  hot,  the  coating  will  wrinkle  on  the  inside  of  the  pipe 
when  it  is  withdrawn  and  laid  aside  to  cool.  The  lengths  of 
pipe  are  put  together  like  stove  pipe,  by  wrapping  a  cloth 
around  the  end  of  one  length  and  driving  it  into  the  end  of 
the  length  below,  the  laying  always  being  started  at  the  lower 
end  of  the  pipe  line.  This  is  called  a  slip  joint.  In  cases 
where  the  pressure  is  considerable  a  sleeve  joint  is  used.  A 
sleeve  joint  consists  of  slipping  an  iron  sleeve  over  the  ends 
where  two  pipe  lengths  join  or  are  butted,  and  running  in 
melted  lead  between  the  sleeve  and  the  pipe,  having  first 
packed  the  joint  sufficiently  to  prevent  the  lead  from  running 
into  the  pipe  where  the  ends  come  together. 

In  rocky,  mountainous  localities  where  trenching  would 
be  quite  expensive,  the  pipe  is  usually  laid  on  the  surface 
without  any  trenching  except  where  it  is  necessary  to  secure 
a  substantial  bearing  or  foundation  for  the  pipe. 

In  very  cold  weather  the  pipe  is  allowed  to  discharge 
constantly,  which  prevents  freezing  within  the  pipe,  or  the 
water  is  prevented  from  entering  the  pipe  and  the  line  left 
empty  when  not  in  use. 

Suppose  a  stream  affords  10  cubic  feet  per  second  and 
has  a  fall  of  400  feet  per  mile,  and  it  is  required  to  construct 
a  water  power  plant  that  will  develop  200  net  horee  power, 
using  a  water  wheel  of  85  per  cent  efficiency.  What  head 
will  be  required  and  what  diameter  and  length  of  pipe,  and 
what  will  be  the  proportions  of  the  discharge  nozzle  required? 

By  formula  (99)  §62,  the  net  head  required  will  be, 
200-=-  85 


SULLIVAN'S  NEW  HYDRAULICS.  181 

As  there  is  a  fall  of  400  feet  per  mile,  it  is  seen  that  the 
line  will  be  a  little  longer  than  oce-half  mile.  The  fall  per 

H  400 

foot  length  will  be  S=  -j-=     528Q  =  .075757575,      and      the 

length  in  which  there  is  a  fall  of  one  foot  is  Z=-~-= —  -  ,—-_ 

=13.20  feet. 

Hence  the  length  of  pipe  required,  not  making  allowance 
for  friction  loss,  will  be  207.065X13.20=2733.258  feet  of  pipe. 

But  as  there  will  be  loss  of  head  by  friction  in  the  pipe 
line  and  also  in  the  nozzle,  and  it  is  required  to  have  207.065 
feet  net  head  at  discharge  from  the  nozzle,  we  must  lengthen 
the  pipe  line  until  the  total  head  will  cover  these  losses  and 
still  leave  the  net  head  of  207.065  feet  at  discharge.  If  the 
nozzle  is  to  be  8  feet  long  we  will  assume  that  the  loss  of 
head  in  the  nozzle  will  be  6  feet  while  discharging  10  cubic 
feet  per  second,  and  we  will  design  the  pipe  line  so  that  the 
loss  of  head  in  the  line  by  friction  will  be  6  feet  also.  Hence 
the  line  must  be  extended  further  down  the  hill  until  we 
have  a  total  head  in  the  whole  length  of  the  line  including 
the  nozzle=207 .065+12=210.065  feet.  In  order  to  gain  this 
additional  12  feet  head  the  line  will  have  to  be  extended  in 
length  by  158.40  feet,  including  the  nozzle.  The  nozzle  is  to 
be  8  feet  in  length,  and  therefore  the  pipe  line  without  the 
nozzle  will  be  (2733.258+158  40)— 8=2883.66  feet  in  length. 
It  is  to  be  double  riveted,  asphaltum  coated,  slip  joiL.t  pipe, 
and  the  total  loss  of  bead  in  the  whole  line  without  the  noz- 
zle is  to  be  6  feet  while  discharging  10  cubic  feet  per  second. 
What  diameter  will  be  required? 

By  formula  (101)  §  64,  the  diameter  required  will  be 
d=U/  .000000269Xq^X^   =1.795  feet=21.54  inches  di- 

ameter. 

We  have  a  net  head  now  at  the  junction  of  the  pipe  line 
with  the  nozzle  of  213.065  feet.  The  next  step  is  to  ascertain 
the  required  dimensions  of  the  nozzle  to  discharge  10  cubic 
feet  per  second  under  these  conditions  with  a  loss  of  head 


182  S'ULLI VAN'S  NEW  HYDRAULICS. 

not  to  exceed  six  feet  in  the  nozzle.  The  method  of  doing 
this  is  explained  in  §65.  For  this  calculation  we  have  the 
length  of  the  nozzle  and  itB  butt  or  greatest  diameter,  and 
the  effective  head  at  the  butt  of  the  nozzle.  Now  if  we  as- 
sume that  there  will  be  probably  a  loss  of  6  feet  head  in  the 
nozzle  itself,  the  net  head  at  discharge  would  be  equal  to  the 
head  at  the  butt,  le-s  the  amount  lost  in  the  nozzle,  or  2 ]  8.065 
—6=207.065  feet.  Hence  by  the  rules  given  heretofore  (§65) 
the  area  in  square  feet  of  the  least  diameter  at  discharge  of 
the  nozzle  will  be 

q  10. 

=-0866  8quare  '66t 


The  diameter  answering  to  this  area  is  A.—-*/— 1 — 

v         .7854 

.11026  foot=1.323  inches 

1.795+.11026 
The  mean  diameter  of  the  nozzle  is= ^ — 

.95263  foot. 

Now  we  must  test  this  nozzle  to  ascertain  what  the  ac- 
tual loss  of  head  will  be  in  it  while  it  is  discharging  10  cubic 
feet  per  second.  If  the  loss  is  not  so  great  as  six  feet,  as  we 
have  assumed  in  the  nozzle,  then  we  mav  shorten  the  pipe 
line  to  some  extent,  or  we  may  reduce  the  diameter  of  the 
pipe  line  very  slightly,  and  still  obtain  the  required  head  and 
power  at  discharge. 

The  velocity  through  the  mean  diameter  of  this  nozzle  in 
order  to  discharge  ten  cubic  feet  per  second  would  be 

q  _  10 

v=~a"  (.95263)"X.7854  =14'03  feet  per  Becond' 

The  actual  loss  of  head  by  friction  in  the  nozzle  under 
these  conditions  would  be  (§  §  37,  39,  65). 

n  1 9  v2  _  .0003623X8X9X ' 96.841 


.9298 


=55224feet. 


As  the  net  head  at  the  base  of  nozzle  is  213.065  feet,  and 
the  loss  in  the  nozzle  is  5.5224    feet,  we  have   a   net  head  at 


SULLIVAN'S  NEW  HYDRAULICS.  183 

point  of  discharge  from  the  nozzle  of  207.5426  feet.  The 
required  net  head  was  207.065  feet.  Hence  we  have 
.4776  foot  head  in  excess  of  exact  requirements,  which 
Is  near  enough  the  desired  result. 

TEST   FOR   QUANTITY   DISCHARGED. 

The  area  of  smallest  diameter  of  nozzle  at  discharge  is 
.0866  square  foot  ae  above  found,  and  the  net  head  at  dis- 
charge ia  207.5426  feet. 

Hence  the  quantity  that  will  be  discharged  is  q=a  v,  or 

q=.OS66X8.0251/207.54  =10.000568  cubic  feet  per  second. 

Now  the  velocity  of  discharge  from  the  nozzle  is  v= 
8.025/26T5T =115.48  feet  per  second  or  115.48X60=  6,928.80 
feet  per  minute. 

It  has  been  established  by  experiment  and  experience 
that  the  velocity  of  greatest  efficiency  of  the  circumference 
of  an  impulse  and  reaction  water  wheel  is  about  one-half  the 
velocity  of  discharge  upon  the  wheel.  The  number  of  revo- 
lutions per  minute  of  the  water  wheel  will  depend  upon  its 
circumference  from  center  to  center  of  the  buckets  taken  as 
its  diameter.  The  circumference  equals  the  diameter  in  feet 
from  center  to  center  of  buckets  multiplied  by  3.1416. 

The  circumference  of  the  wheel  when  the  load  is  on 
should  travel  at  one- half  the  velocity  of  the  discharging 
water.  Hence  the  diameter  of  the  wheel  may  be  so  propor- 
tioned to  the  velocity  of  discharge  as  to  run  any  desired  num- 
ber f  revolutions  per  minute.  Where  high  speed  is  desired 
under  a  low  head,  two  or  more  water  wheels  of  equal  diam- 
eter may  be  placed  upon  one  shaft  and  have  separate  nozzles. 
In  this  way  very  small  diameters  of  the  wheels  may  be  used 
to  secure  high  speed,  and  the  water  divided  so  as  to  avoid 
placing  very  large  buckets  on  small  wheels  and  to  also  pre- 
vent flooding  the  wheel.  The  power  developed  does  not  de- 
pend upon  the  diameter  of  the  water  wheel,  but  depends  up- 
on its  speed  with  reference  to  its  diameter. 

The  point  of  the  nozzle  should  be  firmly  set  beyond  the 
possibility  of  slipping  against  the  wheel,  and  should  be  as 
close  to  the  buckets  as  possible  not  to  strike  them  or  to  have 


184 


SULLIVAN'S  NEW  HYDRAULICS. 


the  jet  re-acted  upon  from  the  buckets.  The  distance  between 
the  point  of  the  nozzle  and  the  center  of  the  bucket  on  the 
wheel  will  depend  upon  the  diameter  at  discharge  of  the  noz- 
zle and  the  velocity  of  discharge  upon  the  wheel.  It  should 
not  be  so  close  that  the  jet  will  react  upon  itself  on  striking 
the  buckets. 

68.—  Table  of  Eleventh  Roots  to  Facilitate  Calcula- 
tions of  Diameter  Required  to  Discharge  Given  Quan- 
tities. 

The  following  table  covers  diameters  from  one  inch  to 
32  inches  both  inclusive,  and  will  be  convenient  in  conjunction 
with  formulas  for  ascertaining  the  diameter  in  feet  required 
to  generate  a  given  discharge  (formulas  28,  43,  65,  81, 100,  101) 
or  to  cause  a  given  discharge  with  a  given  loss  of  head. 
TABLE  No.  36. 


Number 

llth.  Root 

Number 

llth.  Root 

.OOOOOOOOOU01345 
.0000*002762 

.08333 
.1667 

46.24 
86.50 

.417 
.5 

.000000^384 

.25 

156.40 

583 

.000005638 

.3333 

276.20 

.667 

.00006578 

.4167 

471.50 

.75 

.0004883 

.5 

784.80 

.002659 

.5833 

1285.00 

^917 

.01157 

.6667 

2048.00 

2.000 

.04223 

.75 

3203.  00 

2  083 

.1345 

.8333 

4948.00 

2.167 

.3842 

.9167 

7482.00 

2.25 

1.0000 

1.000 

11150.00 

2.404 

1.083 

16370.00 

2^416 

5.467 

1.167 

23840.00 

2  5 

11.64 

1.250 

31300.  (0 

2.584 

23.62 

1.333 

48560.00 

2.667 

REMARK  1.— Where  the  pipe  is  to  be  of  uniform  diameter 
and  to  have  free  discharge,  as  in  the  case  of  a  pipe  conveying 
water  from  one  reservoir  to  another,  there  is  no  object  in  pre- 
serving the  head  by  throttling  the  discharge,  and  in  such  case 
the  total  head  is  consumed  in  balancing  the  resistance  to  flow 
except  that  part  of  the  head  which  is  converted  into  velocity. 
The  diameter  of  a  pipe  which  is  required  to  convey  a  given 
quantity  of  water  per  second  under  such  conditions  will  be 


.3805  H2 


SULLIVAN'S  NEW  HYDRAULICS.  /85 

m=coefficient  of  velocity  in  terms  of  diameter  in  fe*-t. 

q=cubic  feet  per  second  that  pipe  is  to  discharge. 
total  head  in  feet 

S=total  length  in  feet=Bine  of  eloP6 

H=total  head  in  feet. 

REMARK  2—  Where  the  pipe  must  convey  a  given  quantity 
per  second  to  a  given  point  and  must  maintain  a  given  head 
or  pressure  at  that  point  while  the  given  quantity  is  being 
drawn  from  it,  then  the  diameter  required  will  be  found  as 
pointed  out  in  §  G4,  or  by  the  following  general  formula. 


i 

— 


.38u5Xh"8 

This  diameter  will  convey  a  given  quantity  with  a  given 
loss  of  head  which  is  pre-determined  according  to  require- 
ments. 

In  which, 

h"=head  in  feet  to  be  lost  in  friction. 

n=coefficient  of  resistance  applicable  to  class  of  pipe. 

q—  cubic  feet  per  second  pipe  is  to  discharge. 

I  =length  of  pipe  in  feet. 

REMARK  3.—  The  results  of  experiments  by  the  writer  on 
"Converse  Patent  Lock  Joint  Pipe"  made  of  wrought  iron  in 
lengths  of  from  15  to  20  feet  and  lap  welded,  and  coated  with 
asphaltum  gave  an  average  value  of  n=.000299  in  terms  of 
diameter  in  feet.  The  small  value  of  the  coefficient  of  re- 
sistance n,  in  this  pipe  is  to  be  attributed  to  its  uniformity  of 
diameter,  and  to  the  fact  that  it  is  made  in  long  lengths  BO 
there  are  fewer  joints  per  mile  of  pipe,  and  the  joints  are  so 
arranged  as  to  present  a  continuous  and  uniform  surface  to 
the  flow.  For  this  class  of  pipe  take  n=.0003  and  m=:  .00030472 
in  terms  of  diameter  in  feet.  These  values  of  the  coeffi- 
cients do  not  allow  for  future  deposits  in  the  pipe,  if  such 
should  occur,  nor  for  deteriorationinthe  pipe  coating.  It  is  not 
probable  that  a  first  class  asphaltum  coating  will  deteriorate 
to  any  considerable  extent  for  a  great  number  of  years.  This 
remark  has  no  reference  to  coatings  made  of  coal  tar  com- 
pounds. 

The  diameter  (inside)  in  feet  of  Converse  pipe,  asphaltum 
coated,  required  to  convey  a  given  quantity  with  a  given  loss 
of  head,  would  be 


186  SULLIVAN'S  NEW  HYDRAULICS. 

f«  q4 

Or  if  the  discharge  is  to  be  free  and  full  bore,  and  no  at- 
tempt made  to  preserve  the  head  or  pressure,  the  diameter 
required  to  carry  a  given  quantity  will  be 


'a*   m2         11/m2        11 

=  V  »><  l 


11/m2  I2  q4  11  /  Is  q4 

d=T/  :s805^-=-25055i/  -IP- 

ll/  q4 
For  riveted  asphaltum  coated  pipe  d  =.2o41i/          - 

69.— Head  Lost  by  Friction  at  Bends  in   Water  Pipes. 

The  amount  of  the  loss  of  head  produced  by  a  bend  in  a 
pipe  will  depend  upon  the  velocity  of  flow  and  the  radius  of 
the  central  arc  of  the  bend,  and  also  upon  the  number  of  de- 
grees included  in  the  arc- of  the  bend.  Whether  the  addi- 
tional head  required  to  overcome  the  resistance  of  a  bend  will 
be  proportional  to  the  square  or  to  the  cube  of  the  velocity  is 
doubtful.  Weisbach's  formula,  which  is  most  generally  used 
for  determining  the  resistance  of  bends,  gives  results  un- 
doubtedly too  low  in  all  cases  except  for  a  bend  of  90°  with  a 
radius  of  central  arc  of  bend  equal  to  one  half  the  diameter. 

The  resistance  at  a  bend  in  a  pipe  or  in  an  open  channel 
is  caused  by  the  change  of  direction  of  the  flow.  The  more 
abrupt  the  change,  and  the  greater  the  amount  of  the  change 
in  direction,  the  greater  will  the  resistance  be.  It  is  evident 
therefore  that  the  resistance  will  be  directly  as  the  number 
of  degrees  included  in  the  central  arc  of  the  bend  and  in- 
versely as  the  radius  of  that  arc. 


SULLIVAN'S  NEW  HYDRAULICS. 


187 


FIG.B. 


Fig.  A  Bhowe  a  bend  of  90°,  the  radius  c  a,  of  the  central 
arc  of  the  bend  being  equal  6  times  the  radius  a  b,  of  the 
pipe,  or  equal  three  diameters  of  the  pipe.  Fig.  B  shows  a 
bend  of  90°  with  the  radius  c  e  of  the  central  arc  of  the  bend 
equal  the  radius  of  the  pipe,  or  equal  12  diameter  of  pipe. 

When  the  radius  c  a  of  the  central  arc  of  the  bend  is 
only  equal  to  the  radius  of  the  pipe,  or  to  one  half  the  diam- 
eter, then  the  resistance  or  amount  of  head  lost  at  such  bend 
will  equal  the  head  in  feet  which  generates  the  velocity  of 

Vs 

flow,  or  h"=gj.  ±  •    For  example  suppose  the   velocity  to  be  3 

feet  per  second  through  the  pipe,  and  the  bend  is   as  shown 
in  Fig.  B,  then  the  head  in  feet  lost  by  resistance   at  the  bend 

_  (3)2 

~~ 


will  be  h"= 


=  -U  foot 


For  a  bend  of  90°  or  any  other  constant  number  of  de- 
grees, the  amount  of  change  in  the  direction  of  the  flow  will 
be  the  same  for  any  length  of  radius  c  a,  of  the  arc  of  the 
bend,  but  the  distance  in  which  this  change  is  effected  will 
be  directly  as  the  len  th  c  a,  of  the  radius  of  the  bend. 
Hence  the  shorter  the  radius  of  the  bend  the  more 
abrupt  will  be  the  change  in  direction  of  flow,  and  con- 
sequently the  greater  the  resistance.  The  central  arc,  a  ed, 
of  the  bend  increases  in  length  or  becomes  more  gradual 
directly  as  the  radius  of  the  bend  c  a  increases  in  length  and 
hence  the  longer  this  radius  c  a  becomes  the  more  gradual 
will  be  the  change  effected  in  the  direction  of  the  flow.  The 


183  SULLIVAN'S  NEW  HYDRAULICS. 

resistance  at  a  bend  will  therefore  be  directly  as  the  number 
of  degrees  included  by  the  central  arc  of  the  bend,  and  in- 
versely as  the  length  of  the  radius  of  the  bend  and  will  in- 
crease as  v2  (or  possibly  as  v8).  The  formula  will  therefore  be 

,  „  __  Ay,  r  v  va_  A        .5  v    v8       A         .5  y  v2 
~  90  A  R  A  -2g     90  A  R  A  64  4     90  X  64  4  A  R 

Which  reduces  to 


In  this  formula  (102) 
r=$4  diameter  of  pipe=.5 
R=  radius    of    central    arc  of   bend     in  diameters  of 

the    pipe    and    is    to  be  expressed  as  1,  2,  3  etc  di- 

ameters. 
A—  number   of  degrees    of    the  arc  of  bend  as  30,  90, 

180,  etc, 
v=mean  velocity  of  flow  through  the  pipe. 

EXAMPLE   OF   THE    USE   OP    THE   FORMULA. 

It  is  required  to  find  the  resistance  at  a  bend  of  180°  in 
an  eight  inch  pipe  where  the  mean  velocity  is  3  feet  per  sec- 
ond and  the  radius  of  the  central  arc  of  the  bend  is  equal  3 
diameters. 

,.._  A  v*  .007764  _  180X9X.OQ7764  _  n4fifiju  fflflf  head 
90  R  90X3 

REMARK  1.  —  The  resistance  at  a  bend  is  in  addition  to  the 
ordinary  frictional  resistances  of  the  pipe  walls.  Hence  for  a 
pipe  which  contains  a  bend,  first  calculate  the  loss  of  head  by 
friction  as  for  a  straight  pipe,  and  then  add  the  loss  of  head 
due  to  the  bend. 

REMARK  2—  It  is  assumed  in  all  formulas  for  resistance 
at  bends  that  the  resistance  is  independent  of  the  diameter  of 
the  pipe  or  width  of  the  open  channel,  and  that  the  resis- 
tance of  a  bend  depends  solely  upon  the  velocity,  the  radius 
of  the  bend  and  the  number  of  degrees  included  in  the 
central  arc  of  the  bend.  It  is  doubtful  whether  the  diameter 
of  a  pipe  exerts  an  influence  on  the  resistance  at  a  bend  or 
not.  It  probably  does. 


SULLIVAN'S  NEW  HYDRAULICS.  189 

REMARK  3 — The  force  exerted  by  a  column  of  water  im- 
pinging upon  a  fixed  surface  is  as  the  product  of  the  quantity 
of  water  by  its  head.  The  quantity  ip  directly  as  the  velocity 
and  the  head  is  as  the  square  of  the  velocity.  Consequently 
the  product  is  vXv8=v3.  It  is  therefore  possible  that  the 
force  or  head  or  energy  absorbed  at  a  bend  will  vary  as  v»  in- 
stead of  v8. 

70— Formulas  of  Weisbach  and  of  Rankine  for  Resis 
tance  at  Bends  in  Pipes. 

The  formula  for  resistance  at  bends  proposed  by  Weis- 
bach is 

h'=.l31+1.847  (-LVx-^-X— 

In  which 

r=radius  of  pipe  in  feet=^  diameter  in  feet. 

R=radius  of  axis  of  bend  in  feet. 

A=central  angle  of  bend  in  degrees. 

2g=effect  of  gravity=64.4. 

Professor  W.  J.  M.  Rankine's  formula  is  simply  a  change 
in  form  of  Weisbach's  formula,  and  is  as  follows: 


In  which 

A=angle  of  bend  in  degrees 

d=diameter  in  feet  of  pipe 

r=radius  of  central  arc  of  bend 
To  simplify   Weisbach's   formula,  place   the  coefficient, 

.131+1.847(-^-)*=Z.    Then 
V  R  / 

,  „     „.,  A  ^,  v8        AvaZ       y  /   A  v2  \ 
=ZXT80X  60=-11592-=  Z\TI592  ) 

Remembering  that  in  Weisbach's  formula,  r=  half  the 
diameter  of  the  pipe  in  feet,  and  R=  radius  of  the  central  arc 
of  the  bend  in  feet,  the  following  table  of  values  of  Z  will  be 
readily  understood  and  applied:- 


190  SULLIVAN'S  NEW  HYDRAULICS. 

Value  of  Z  in  Weisbach's  Formula. 


r 

.1 

.15 

.2 

.225 

.25 

.275 

.3 

.325 

.35 

.375 

.4 

.425 

Z= 

.131 

.133 

.138 

.145 

.15 

.155 

.16 

.17 

.18 

.195 

,20i  ; 

.225 

r 

.45 

.475 

.5 

.525 

.55 

.575 

.6 

.625 

.65 

.675 

.7 

.725 

Z= 

.244 

.264 

.294 

.32 

.35 

.39 

.44 

.49 

.54 

.60 

.661 

.73 

r 

.75 

.775 

.80 

.825 

.85 

.875 

.9 

.925 

.95 

.975 

l.(X) 

Z= 

.806 

.880 

.98 

1.0- 

Ujj 

1.  29L  41 

1.51 

1.68ll.8:r2.00l 

USE  OP  ABOVE  TABLE.  —  The  velocity  iu  feet  per  second 
through  an  eight  inch  pipe  is  3  feet.  There  is  a  bend  of  90° 
with  a  radius  of  bend  equal  4  inches  or  half  the  diameter. 
What  is  the  loss  of  head  in  feet  caused  by  the  bend? 

We  see  that  as  the  radius  of  the  central  arc  of   the  bend 

is  equal  to  half  the  diameter  of  the  pipe;  that    -D-=1.00.    Re- 

ferring to  the  above  table,  and  it  is  seen  that  when^s-  =  1.00, 
then  Z=2.00.    Hence  by  the  formula, 


h"=z- 


The  radius  in  feet  of  an  8  inch  pipe=.3333  foot. 
The  diameter  in  feet  of  an  8  inch  pipe=.6666foot. 
Suppose  the  radius  of  the  above  bend  R=.66G6  foot  or 
equal  the  diameter,  and  the  radius  of  the  pipe  is  .3333  foot. 


-    .6666  ~-5 


From  the  above  table  it  is  seen  that  when-p~  =  -5,  then 
Z=.294.  And  in  this  case  Weisbach's  formula  would  give  the 
loss  for  3  feet  velocity  of  flow  as 


=.^[ggf-]==.(*feet 


h"=Z    111592 

This  latter  result  is  altogether  too  small. 


SULLIVAN'S  NEW  HYDRAULICS. 


191 


71.  —  Comparison  of  the  Results  by  Weisbach's  Formula 
and  by  the  Formula  Herein  proposed,  for  Bends  of  90°  with 
Radii  Varying  form  /?=  yzd  t o  R= 3d,  and  Different  Veloci- 
ties. 

In  the  following  table  the  lose  of  head  by  friction  has  been 
computed  by  our  formula  (102)  and  also  by  Weisbach's  for- 
mula for  various  velocities  of  flow  through  a  bend  of  90°  in 
which  the  radius  of  the  central  arc  of  the  bend  varies  from  R 
=^  d  to  R=3d.  It  is  possible  that  the  results  by  either  for- 
mula are  too  small  for  the  reason  suggested  in  remark  3,  §  69 

TABLE  No.  38. 
Table  of  computed  results  for  comparison. 


2 

g 

4 

5 

6 

7 

8 

Velocities 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

Feet 

.062 

.14 

.248 

388 

56 

.761 

.994 

(  Formula   (102) 

.0621 

.14 

.248 

.388 

.56 

.761 

.994 

1  Weisbach 

fl—d  •< 

.031 
.009 

.07 
02 

.124 
.026 

.198 
.056 

.28 
081 

.38 
.11 

.497 
.144 

i  Formula    (102) 
1  Weisbach 

R=2d  \ 

.0155 
.005 

.035 
.011 

.072 
.(19 

.C97 
.029 

.14 

.042 

.19 
.057 

.248 
.074 

j  Formula   (102) 
1  Weisbach 

j 

.0103 

0233 

.0413 

.'646 

.093 

.1656 

j  Formula   (102) 

.001 

.01 

.017 

.027 

.038 

.052 

.069 

1  Weisbach 

It  would  appear  from  au  inspection  of  the  results  by 
Weisbach's  formula  that  there  is  little  to  be  gained  by  mak- 
ing the  radius  of  the  bend  greater  Iban  twice  the  diameter  of 
the  pipe.  This  is  not  true,  however,  in  practice.  The  radius 
of  a  becd  should  be  made  as  great  as  the  circumstances  will 
permit  unless  the  velocity  of  flow  through  the  pipe  is  to  be 
very  small.  The  velocity  should  be  the  controlling  feature 
in  determining  the  radius  of  the  bend. 

Fanning  says  ''Our  bends  should  have  a  radius,  at  axis, 
equal  at  least  to  4  diameters."  Trautwine  advises  a  radius  of 
bend  equal  to  5  diameters  length,  or  as  much  longer  as  it  can 
be  made.  If  the  velocity  does  not  exceed  5  feet  per  second, 
then  a  radius  of  3  diameters  will  reduce  the  loss  of  head  to 
.0646  foot  at  a  90°  bend. 


192  SULLIVAN'S  NEW  HYDRAULICS. 

72.— Resistance  at  Bends.    Rennie's  Fxperiments. 

While  the  results  of  experiments  by  Bennie  on  leaden 
pipe  one  half  inch  diameter  are  not  of  great  value  as  estab- 
lishing any  law  of  resistance  at  bends,  yet  they  indicate 
very  clearly  that  the  results  by  Weisbach's  formula  are  too 
low. 

Bennie  experimented  with  a  leaden  pipe  15  feet  in  length 
and  half  inch  in  diameter  under  a  total  head  of  4  feet.  He  ob. 
tained  the  following  results; 

The  straight  pipe  before  being  bent  discharged  .00699 
cubic  feet  per  second. 

With  one  bend  at  right  angles  near  the  end, 00556 

cubic  feet  per  second. 

With  24  right  angle  bends 00253 

cubic  feet  per  second. 

It  will  be  noted  that  the  bends  are  described  as  right 
angled.  This  may  have  crushed  the  pipe  out  of  form  and  re- 
duced the  area  at  the  bends,  This  would  materially  affect  the 
velocity  and  the  resistance  through  the  bend.  Whether  this 
occured  or  not  is  not  stated,  Prom  the  area  in  square  feet 
of  this  half  inch  pipe  and  the  quantity  in  cubic  feet  per  sec- 
ond it  discharged  we  find  that  the  velocities  of  discharge  were 
as  follows: 

Before  the  pipe  was  bent,  v  =  JL=  •°06^9  =5  feet  per  sec- 
ond. 

With  one  right  angle  bend,  v  =-S_=  -00556  =  3.971  feet 
per  second. 

With  24  right  angled  bends,  v=-i-=-^HL=  1.80    feet 

per  second. 

In  order  to  prevent  confusing  the  resistance  of  the  pipe 
walls  with  that  of  the  bends,  we  will  first  find  the  value  of 
the  coefficient  of  resistance  n,  of  the  pipe  before  it  was  bent. 

The  total  head  was  4  feet,  and  while  the  pipe  was  straight 
the  velocity  of  discharge  was  5  feet  per  eecond.  The  head  in 
feet  lost  by  friction  along  the  walls  of  the  straight  pipe  under 
this  velocity  was  equal  the  total  head  minus  the  head  due  to 
the  velocity  of  discharge,  or  was 


SULLIVAN'S  NEW  HYDRAULICS.  193 


4  —  ^f=4.—  .3882=3.6118  feet. 

After  one  bend  was  made  in  the  pipe,  the  total  head  re- 
maining 4:  feet,  the  velocity  of  discharge  was  only  3.971  feet 
per  second.  Now  from  the  data  of  flow  in  the  straight  pipe 
before  the  bend  was  introduced  we  find  the  value  of  D  to  be 

=    Jv*  15X25    =-000082- 

After  one  bend  had  been  introduced  the  velocity  was  re- 
duced to  3.971  feet  per  second,  so  the  friction  of  the  pipe  walls 
exclusive  of  the  resistance  of  the  bend  was  now 

nlv"_.000082Xl5X15.76884 
h=-/oT-  .0085 

But  the  total  loss  of  head  due  to  pipe  walls  and  one  bend 
combined  was  equal  the  total  head  of  4  feet  minus  the  ve- 
locity head,  or  equal 
/o  971  \» 

4        644    =*•—  .244842=3.755158  feet. 

If  we  deduct  from  this  total  loss  the  loss  due  to  pipe 
walls  we  have  3.755158—2.2818=1.473358  feet  head  lost  by  the 
resistance  at  the  bend;  which  is  equal  6  times  the  head  gen- 
erating the  velocity.  This  would  indicate  that  the  resistance 
at  a  bend  is  more  nearly  proportional  to  v*  than  to  v*.  as  inti- 
mated in  remark  3,  §  69.  The  resistance  at  a  bend  in  a  very 
small  pipe  is  probably  greater  than  in  large  pipes, 

The  total  head  remaining  4  feet,  after  24  right  angled 
bends  were  made  in  this  15  foot  length  of  half  inch  lead  pipe 
the  velocity  was  1.80  feet  per  second  as  determined  from  the 
quantity  discharged.  The  loss  of  head  due  to  friction  of  pipe 
walls,  exclusive  of  the  bends,  was,  for  this  velocity. 

h,=  nZv«  =  .  000082X15X3.24  =  mB  feet  head> 
v'd3  .0085 

The    total   loss  of  head  due  both  to  the  24  bends  and  the 

friction  of  pipe  wall  wae  H—_  lL=  4.—  .05031=3.94969  feet. 

•*& 

The  loss  due  to  the  24  bends  alone  was    therefore  equal 


194  SULLIVAN'S  NEW  HYDRAULICS. 

the  total  loss  minus  the  loss  due  to  pipe  walls=3.94969— .4688 
=3.48  feet. 

If    the  loss   was   equal   at   each  bend,  then  h°=  3J48  = 

.145  foot  head  lost  at  each  bend  for  a  velocity  of  1.80  feet  per 
second.  In  this  caee  the  head  lost  at  each  bend  was  only 
equal  2.88  times  the  head  generating  the  velocity.  It  must 
be  remembered  that  all  these  bends  are  described  as  right 
angled  bends,  It  is  probable  that  serious  contractions  of  the 
area  of  the  pipe  were  produced  at  each  such  bend  and  that 
the  velocity  of  flow  through  the  contractions  was  greater 
than  1.80  feet  per  second. 

Because  of  the  direct  action  and  equal  reaction  of  the 
water  impinging  upon  the  pipe  wall  at  a  right  angled  bend 
the  lose  of  head  at  such  bend  could  not  be  less  than  twice 

v2 
the  head  producing  velocity,  or  h"  =  2-sr-     According  to  the 

above  results  of  Ronnie's  experiments  with  24  right  angled 
bends  it  appears  that  the  loss  at  each  bend  was  equal  nearly 

three  times  the  head  producing  the  velocity  or  h"=2.88  -| — 

But  it  is  doubtful  what  the  actual  velocity  was  in  the  bends 
as  the  areas  were  probably  contracted. 

Right  angled  bends  or  shoulders  are,  however,  never  in- 
troduced into  a  water  pipe,  but  the  bends  are  always  circular. 
As  a  true  right  angled  bend  cannot  be  made  without  cutting 
and  fitting,  or  casting,  it  is  probable  that  Rennie's  pipe  was 
bent  like  Fig.  B,  §69. 

73.— Relation  of  Thickness  of  Pipe  Shell  to  Pressure, 
Diameter  and  Tensile  Strength  of  Pipe  Metal 

When  a  pipe  is  filled  with  water  and  is  closed  at  dis- 
charge end  so  there  can  be  no  flow  in  it,  the  radial  pressure 
within  the  pipe  tending  to  burst  it  will  vary  as  the  head  of 
water  above  any  given  point  along  the  pipe,  and  at  any 


SULLIVAN'S  NEW  HYDRAULICS. 


195 


given  point  will  be  equal  HX.434=lbs.  pressure    on  each 
square  inch  of  the  internal  circumference. 
A  E  F 


In  the  Figure  let  R  represent  a  reservoir,  the  water  level 
in  which  is  A,  and  a  pipe  C  D  G,  is  laid  from  it  ov*r  hills  and 
depressions.  When  the  pipe  is  closed  at  G,  the  pressure 
within  the  pipe  which  tends  to  burst  it  will  vary  as  the  ver- 
tical distance  C  E,  D  F,  between  the  given  point  in  the  pipe 
and  the  level  of  the  water  A  E  F  B,  in  the  reservoir.  Hence 
the  thickness  and  strength  of  the  pipe  shell  must  be  pro- 
portion according  to  the  position  it  is  to  occupy  in  the  pipe 
line.  If  the  vertical  distance  CE  is  J30  feet  then  the  pressure 
at  C  on  each  squara  inch  of  the  internal  circumference  of  the 
pipe  will  be  130X .434=56.42  Ibs.  But  the  pipe  passing  over  the 
hill  at  D  is  only  80  feet  below  the  level  of  the  water  in  the 
reservoir,  and  consequently  the  pressure  within  the  pipe  at  D 
is  equal  80X.434=34.7'2  Ibs.  per  square  inch.  A  profile  of  the 
pipe  line  showing  the  distance  at  all  rises  and  depressions 
along  the  line  between  the  pipe  and  the  level  A  E  F  B  should 
always  be  made  before  the  thickness  of  pipe  shell  is  calcula- 
ted for  any  portion  of  the  line.  With  such  profile  the  thick- 
ness and  strength  of  the  pipe  for  each  division  of  the  line 
may  be  calculated  so  as  to  conform  to  the  pressure  it  must 
sustain. 

The  inclined  line  A,  G,  is  the  hydraulic  grade  line,  or  line 
which  indicates  the  hydraulic  or  running  pressure  in  the 
pipe  when  the  pipe  is  open  at  G  and  discharging  freely. 

The  hydraulic  or  running  pressure  within  the  pipe  at  any 
given  point  along  the  pipe  line  is  equal  to  the  distance  in- 
feet,  measured  vertically,  from  the  given  point  in  the  pipe  to 


196  SULLIVAN'S  NEW  HYDRAULICS. 

the  hydraulic  grade  line,  A  G.,  multiplied  by  .431.  Thus,  the 
running  pressure  at  C  in  the  pipe  ia  equal  the  vertical  dis- 
tance C  H  in  feet  multiplied  by  .434.  The  difference  in  feet 
between  C  E  and  C  H  shows  the  loss  of  head  in  feet  by  fric- 
tion between  the  reservoir  and  C.  If  the  pipe  were  laid  on 
the  hydraulic  grade  line  A,  G,  there  would  be  no  pressure  in 
it  at  all  when  discharging  freely  except  that  due  to  the  depth 
of  the  diameter.  The  pipe  must  be  so  laid  that  no  part  of  it 
will  rise  above  the  hydraulic  grade  line.  If  the  pipe  at  D 
should  rise  above  the  line  A  G,  to  K,  then  the  line  would  re- 
quire to  be  divided  into  two  divisions,  A  K,  and  K  G,  both  as 
to  diameter  of  pipe  and  as  to  the  hydraulic  grade  line.  The 
diameter  KG,  if  the  same  as  A  K,  would  not  run  full,  for  the 
reason  that  K  G  would  have  a  greater  fall  per  foot  length 
than  A  K.  Assuming  the  pipe  to  be  laid  as  shown  by  C  D  G, 
and  that  it  is  closed  at  G  so  there  is  no  discharge,  then  the 
internal  pressure  on  each  square  inch  at  any  given  point  in 
the  pipe  will  equal  the  vertical  head  in  feet  between  the 
given  point  in  the|pipe  and  the  line  A  E  P  B,  multiplied  by  .434, 
and  the  number  of  square  inches  subject  to  this  pressure  will 
be  directly  as  the  diameter  in  inches  of  the  pipe,  because  the 
circumference  is  equal  dX3.1416. 

The  total  pressure  on  the  inner  circumference  will  there- 
fore be  HX.434XdX3.U16. 

The  pressure  of  quiet  water  is  equal  in  all  directions.  In 
a  circular  pipe  the  pressure  radiates  from  the  axis  of  the  pipe 
to  every  point  in  the  circumference.  The  resultant  of  the 
pressure  on  one  half  the  circumference  acts  through  the 
center  of  gravity  of  that  half,  and  equals  the  products  of  the 
pressure  into  the  projection  ofjthat  half  circumference.  The 
projection  of  half  the  circumference  equals  the  diameter  of 
the  pipe.  An  equal  resultant  acts  in  the  opposite  direction 
through  the  center  of  gravity  of  the  other  half  circumference. 
The  resulting  strain  on  the  pipe  shell  at  any  point  in  the  cir- 
cumference is  equal  to  the  sum  of  these  opposing  resultants. 
If  therefore,  the  thickness  and  strength  of  the  pipe  shell  is  to 
be  lound  simply  in  terms  of  the  pressure  resultant  of  one  half 


SULLIVAN'oQ  NEW  HYDRAULICS.  197 

the  circumference,  due  to  the  total  head,  it  is  evident  that  the 
thickness  and  strength  must  equal  twice  this  resultant,  or, 
2tS=PXd (103) 

t=  thickness  of  pipe  shell  in  inches. 

S=  tensile  strength  in  Ibs.  per  square  inch  of  pipe  metals. 

P=  pressure  in  Ibs.  per  square  inch  -  H.X-434. 

d=  inside  diameter  of  pipe  ininches. 

This  gives  a  thickness  and  strength  just  sufficient  to 
equal  or  balance  the  pressure  of  the  quiet  water,  as 

t— ^g- OM) 

To  be  sufficiently  strong  to  withstand  the  violent  shocks 
and  sudden  strains  caused  by  water  ram,  and  to  provide  for 
defects  in  casting  or  in  riveting,  and  to  prevent  breakage  in 
handling  and  from  unequal  settlement  of  the  pipe  in  the 
trench,  it  is  necessary  to  make  cast  iron  pipe  very  much 
thicker  and  heavier  than  theory  would  indicate,  and  wrought 
iron  and  steel  pipe  from  three  to  six  times  as  thick  as  the 
quiet  pressure  alone  would  actually  require.  For  these  rea- 
sons the  formula  (104)  must  have  added  to  it  another  factor 
called  the  factor  of  safety,  and  it  then  becomes 

*  —   Pd-XF (105) 


2S 

The  factor  of  safety  F,  may  be  equal  2,  3,  4  etc.  according 
to  the  service  the  wrought  iron  or  steel  pipe  is  to  be  put  to. 

This  formula  is  not  used  for  cast  iron  pipe  for  the  reason 
that  cast  iron  pipe  is  BO  brittle  that  it  is  necessary  to  give  it 
heavy  dimensions  regardless  of  the  pressure  it  is  to  with- 
stand, Wrought  iron  and  steel  pipe  being  flexible  and 
tough,  does  not  require  high  factors  of  safety,  but  if  laid  as 
a  permanent  line,  the  shell  should  be  sufficiently  thick  to 
prevent  pitting  through  in  case  the  coating  is  knocked  off. 
The  factor  of  safety  of  a  pipe  is  found  by  the  formula 


(106). 


The  value  of  S  depends  on  the  net  strength  of  a  riveted 
joint,  (See  §  74) 


198  SULLIVAN'S  NEW  HYDRAULICS. 

Many  steel  pipes  have  been  in  successful  use  under  high 
pressure  for  many  years  with  factors  of  safety  as  low  as  2. 
These  small  factors  of  safety  were  used,  however,  where  the 
pipe  was  not  subject  to  water  ram. 

For  the  reason  heretofore  mentioned,  the  formulas  for 
the  thickness  of  cast  iron  pipe  are  necessarily  arbitrary  and 
empirical. 

For  thickness  in  inches'of  cast  iron  pipe  of  diameters  of 
less  than  60  inches 

t=(P+100)X.OOOH2Xd+.33(l.— .01  d) 

For  thickness  in  inches  of  cast  iron  pipe  of  60  inches  di- 
ameter or  greater, 

t=(P-r-100)X-OOOU2Xd. 

t=thickness  of  pipe  shell  in  inches. 

P=pressure  in  pounds  per  square  inch. 

d=diameter  (inside)  of  pipe  in  inches. 

The  tensile  strength  of  cast  iron  pipe  is  ordinarily  taken 
as  equal  to  18,000  pounds  per  square  inch.  If  made  of  the 
best  quality  of  iron  and  remelted  four  times,  and  cast  verti- 
cally with  bell  end  down,  the  pipe  would  have  a  tensile 
strength  as  great  as  30,000  pounds  per  square  inch,  and  would 
be  tough,  so  that  a  large  part  of  its  superfluous  weight  might 
be  dispensed  with,  and  the  thickness  of  shell  greatly  reduced 
thus  reducing  the  cost  of  freight,  hauling  and  laying. 

74.— Values  of  S  in  Water  Pipe~tFormulas.—The  value 
of  S  to  be  used  in  the  formula  (105)  for  determining  the  re- 
quired thickness  and  strength  of  pipe  shell  depends  on  the 
the  nature  of  the  pipe,  whether  steel  or  iron,  and  whether 
welded  or  riveted,  and  if  riveted,  then  whether  single  or 
double  riveted.  The  net  strength  of  a  riveted  joint  depends 
on  the  ratio  of  shearing  strength  of  rivets  to  tensile  strength 
of  the  plate,  and  also  upon  whether  the  riveting  is  done  by 
hand  or  by  hydraulic  power.  In  hand  riveting  the  work  is  done 
with  cold  rivets  and  the  rivet  boles  are  made  from  1-32  to  1-16 
jnch  larger  than  the  diameter  of  the  rivet,  and  the  effect  of  the 


SULLIVAN'S  NEW  HYDRAULICS.  199 

hammer  in  upsetting  the  rivet  is  not  sufficient  to  swell  the 
rivet  to  its  full  length  so  as  to  completely  fill  the  rivet  hole. 
Hand  riveting  does  not  leave  as  substantial  a  head  on  the 
rivet  as  machine  riveting  and  is  inferior  to  machine  riveting 
in  many  respects.  A  formula  for  fixing  the  pitch  of  rivets  in 
a  joint  is  necessarily  based  on  the  ratio  of  the  given  shearing 
strength  per  square  inch  of  the  rivet  metal  to  the  given  ten 
sile  strength  of  the  plate  metal.  The  formula  must  be  varied 
at  these  factors  vary.  The  tensile  strength  of  wrought  iron 
plates  varies  from  44,000  to  57,000  Ibs  per  square  inch.  A 
good  average  wrought  iron  plate  should  have  a  tensile 
strength  of  50,000  pounds  per  square  inch  before  the  rivet 
holee  are  made  in  it.  The  tensile  strength  of  solid  steel  plate 
varies  from  56,000  to  108,000  Ibs  per  square  inch. 

The  best  iron  rivets  have  a  shearing  strength  of  only 
45,000  Ibs.  per  square  inch.  The  results  of  a  great  many  ex- 
periments made  by  the  Research  Committee  of  the  Institu- 
tion of  Mechanical  Engineers  (London,  1881)  showed  that  the 
ultimate  shearing  resistance  of  steel  rivets  was  49,280  Ibs. 
per  square  inch  for  single  riveted  joints,  and  53,760  Ibs.  per 
square  inch  for  double  riveted  lap  joints.  It  is  very  proba- 
ble that  iron  rivets  would  not  have  a  greater  ultimate  shear- 
ing resistance  than  40,000  Ibs.  per  square  inch  of  livet  area  in 
a  single  riveted  joint  riveted  by  hand.  Very  high  steel  of 
great  shearing  strength  is  too  brittle  for  rivets,  although  riv- 
eted hot.  Hence  there  is  no  advantage  in  adopting  plates  of 
greater  tensile  strength  than  rivets  of  suitable  shearing 
strength  can  be  found  for.  A  steel  plate  of  about  66,000  to 
70,000  Ibs.  per  square  inch  tensile  strength  is  as  high  as  suit- 
able rivets  can  be  obtained  for,  and  plates  of  this  class  will 
require  steel  rivets  of  best  quality.  The  value  of  S  to  be 
used  in  the  formula  (105)  should  be  the  net  strength  of  the 
joint  or  pipe  shell.  We  will  first  give  the  formula  for  propor- 
tions of  riveted  joints,  and  then  for  testing  the  strength  of 
such  joints.  By  these 'means  the  value  of  S  must  he  deter- 
mined in  each  case.  (See  §  §  75  80.) 


200  SULLIVAN'S  NEW  HYDRAULICS. 

75.— Riveted  Steel  Pipe  —For  riveting  cold,  the  best 
grade  of  steel  plate  is  open  hearth  mild  steel  of  about  60,000 
Ibs.  per  square  inch  tensile  strength  to  be  riveted  with  best 
quality  swede  iron  rivets  of  45,000  Ibs,  per  square  inch  shear- 
resistance.  We  have  then  4500°  =75  per  cent  as  the  ratio 
oOOOO 

of  shearing  strength  of  rivets  to  tensile  strength  of  plates.  In 
this  case  1  =1.33,  is  the  ratio  of  area  of  rivets  to  net  plate 

required  to  balance  the  tensile  strength  of  the  plate.  When 
the  rivet  holes  are  made  in  the  plate  it  is  weakened  as  a  whole 
by  a  percentage  found  thus: 

Let  S=Original  tensile  strength  :>f  plate,  unperforated. 

S'=tensile  strength  of  plate  after  holes  are  made. 

P=pitch,  inches,  center  to    center  of  rivets  In  one 
row. 

d=diameter  in  inches  of  rivet  hole  (not  of  rivet). 

t=  thickness  of  plate  in  inches. 

Then  the  per  cent  strength  of  the    punched  plate  S' ,  to 
the  original  unpunched  plate  will  be 

S'  =  P~d  =per  centS 

The  numerical  value  of  S'  will 

r    p      j    -\ 
Q'      "I     *• — *-*     I  v/Qx/4 

O    =       ?s X&XI 


We  have  just  seen  that  in  order  to  make   the  shearing 
•  strength  of  the  rivets  equal  to  the  tensile  strengh  of  the  plate 
;  in  this  case,  the  combined  area  of  the  rivets  must  equal   1.33 
times  the  net  plate  area  between  holes.     The  plate   area  be- 
tween the  rivets  holes  is 
(P-d)t 

The  area  of  the  rivets  is  d8X. 7854.     Hence  the  equation 
(P— d)Xt  X  1.33=d*X  .7854 
From  which 


SULLIVAN'S  NEW  HYDRAULICS.  201 

And 
p=      d».7854x2      +d=L20  -^L+d,  for  double  riveted 

joint, 

But  eupose  the  rivets  had  been  steel  rivets  of  50,000  plbs. 
shearing  strength,  and  the  plates  as  above,  that  is,  of  60,000 
Ibs.  per  square  inch  tensile  strength.  Then  the  pitch  formu- 
la would  be  worked  out  as  follows: 

50,000    Ibs.  shearing  strength 

60,000    Ibs.    tensile    strength  ='833  Per 

Hence,-  030=  1-20.    That  is,  the  combined  area  of  rivets 

must  be  1.20  times  the  net  plate  area  between  holes. 
Then, 

(P_d)tXl.20=d*  .7854 
Prom  which, 

p=  [  dtxi82o  ]  +  d=-6545  -r~+d> for  Bin*le  riveted 

And 

d=1.3094-+d,  for  double  riveted 

Observe  that  d=diameter  of  rivet  hole,  which  is  always 
from  1  32  to  1-16  inch  larger  than  the  rivet  before  the  rivet 
is  upset. 

We  are  restricted  to  the  use  of  the  market  sizes  of  rivets, 
and  should  select  a  diameter  of  rivet  equal  to  from  1.70  to  2.33 
thicknesses  of  the  plate.  When  the  diameter  of  rivet  is  select- 
ed then  add  1-32  (.03125  inch)  for  value  of  d  in  the  pitch  formula 

If  steel  plate  of  70,000  Ibs.  per  square  inch  tensile 
strength  is  used,  then  the  best  quality  of  steel  rivets  of  not 
less  than  53,000  Ibs.  per  square  inch  shearing  resistance 
should  be  adopted.  In  this  case  the  combined  area  of  the 
rivets  must  exceed  the  area  of  the  net  plate  metal  between 
rivet  holes  by  .32075  per  cent,  as  below  shown. 
70000 

'53000~=  per 


202  SULLIVAN'S  NEW  HYDRAULICS. 

Then, 

(P—  d)tX  1.32075=d2  .7854=total  area  of  rivets. 

And, 


itXl.320?5  ]    +d=-5946-^+d,  for  single  riveted 


joints 
And 


P=  +d=1-19-+d'    for    double   riveted 


joints. 

If  the  pipe  is  to  sustain  an  extremely  high  pressure,  or  is 
subject  to  frequent  water  ram,  it  should  be  triple  riveted 
with  a  ribbon  of  lead  1-32  inch  thick  placed  between  the  lap 
of  the  plates.  Then  for  a  triple  riveted  joint  with  rivets  and 
plates  of  the  above  strengths,  the  pitch  formula  would  be 

P=    [tx  1.32075  J    +d=1.784—  +d,  center  to  center,  in 

one  row. 

After  many  tests  of  riveted  joints  (steel  plates  and  steel 
rivets)  the  Research  Committee  of  the  Institution  of  Mechan- 
ical Engineers  (London,  1881)  reported  that:  "To  attain  the 
maximum  strength  of  joint  the  breadth  of  lap  must  be  such 
as  to  prevent  it  from  breaking  zig-zag.  It  has  been 
found  that  the  net  metal  measured  zig-zag  should  be 
from  30  to  35  per  cent  in  excess  of  that  measured  straight 
across,  »JQ  order  to  insure  a  straight  fracture.  This  corres- 

2  d 

ponds  to  a   diagonal   pitch   of  -5-  P  -f-~  o~»  if  P  be  the  straight 

pitch  and  d=diameter  of  rivet  hole  To  find  the  proper 
breadth  of  lap  for  a  double  riveted  joint  it  is  probably  best 
to  proceed  by  first  setting  this  pitch  off,  and  then  finding 
from  it  the  longitudinal  pitch,  or  distance  between  the  cen- 
ters of  the  two  rivet  lines  running  parallel  across  the  plate." 
If  the  net  metal  between  two  rows  of  rivet  holes  is  equal 
to  twice  the  diameter  of  the  rivet  hole,  the  joint  will  be  safe. 


SULLIVAN'S  NEW  HYDRAULICS.  203 

The  distance  of  the  rivet  holes  from  edge  of  plate  should  be 
equal  to  two  diameters  of  the  rivet  hole. 

In  the  experiments  of  the  Research  Committee  they 
found  that  a  single  riveted  joint,  riveted  by  hand,  (steel  rivets 
and  plate)  would  begin  to  slip  or  give  when  the  stress  or  load 
per  rivet  amounted  to  6,600  Ibs.  The  plates  were  3-8  inch 
thick  and  rivets  one  inch  diameter.  A  similar  hand  riveted, 
double  riveted  joint,  began  to  slip  or  give  when  the  load  per 
rivet  reached  7,840  Ibs.  whereas  a  machine  riveted  joint  of 
similar  proportions  did  not  begin  to  slip  until  the  load  per 
rivet  was  double  that  at  which  the  hand  riveted  joints  began 
to  give. 

The  value  of  hydraulic  riveting  is  in  the  fact  that.it  holds 
the  plates  more  tightly  together,  and  thus  doubles  the  load 
at  which  the  slip  in  a  joint  commences.  The  size  of  rivet 
heads  and  ends  was  found  of  great  importance  in  single 
riveted  joints.  An  increase  of  one-third  in  the  weight  of 
the  rivets  (all  the  excess  weight  being  in  the  rivet  heads  and 
ends)  was  found  to  add  8  1-2  per  cent  to  the  resistance  of  the 
joint,  for  the  reason  that  the  large  heads  and  ends  held  the 
plates  firmly  together  and  prevented  them  from  cocking  so  as 
to  place  a  tensile  strain  on  the  rivets.  The  committee  also 
found  that  the  effect  of  punching  instead  of  drilling  the  rivet 
holes  was  to  weaken  the  plates  from  5  to  10  per  cent  in  soft 
wrought  iron,  and  20  to  25  per  cent  in  hard  wrought  iron 
plates,  and  20  to  28  per  cent  in  steel  plates.  This  weakening, 
of  coursrf,  extended  only  to  the  metal  immediately  around  the 
hole.  They  also  found  that  the  metal  between  the  rivet  holes 
in  mild  steel  plate  has  a  considerably  greater  tensile  strength 
per  square  inch  than  the  unperforated  metal.  The  excess 
tensile  strength  amounted  to  from  8  to  20  per  cent,  being 
largest  where  the  distance  between  rivet  holes  was  least. 

"A  riveted  joint  may  yield  in  three  ways  after  being 
properly  proportioned,  namely,  by  the  shearing  of  its  rivets; 
or  by  the  pulling  apart  of  the  net  plate  between  the  rive* 
holes;  or  by  the  crippling  (a  kind  of  compression,  mashing  or 
crumpling)  of  the  plates  by  the  rivets  when  the  two  are  too 


204  SULLIVAN'S  NEW  HYDRAULICS. 

forcibly  pulled  against  each  other.  It  also  compresses  the 
rivets  themselves  transversely  at  a  less  strain  than  a  shearing 
one;  and  this  partial  yielding  of  both  plates  and  rivets  al- 
lows the  joint  to  stretch  considerably  before  there  is  any 
danger  of  actual  fracture.  Or  in  steam  or  water  joints  it  may 
cause  leaks  without  further  inconvenience  or  danger." — 
Trautwine. 

In  view  of  the  results  of  the  experiments  as  to  the  slip- 
ping, or  giving  or  "crippling"  of  joints,  as  shown  by  the  re 
port  of  the  Research  Committee,  it  is  evident  that  if  an  ab 
solutely  water  tight  joint  is  to  be  made  to  stand  high  pres- 
sure, the  pitch  of  the  rivets  must  be  less  than  would  be  in- 
dicated by  the  theory  of  simply  equalizing  the  shearing 
strength  of  rivets  and  the  tensile  strength  of  plates.  The 
crushing  or  mashing  load,  within  elastic,  limits,  must  be 
observed. 

76.— Table  ot  Proportions  of  Single  and  Double  Riv- 
eted Joint,  Mild  Steel,  Water  Pipe  Joints.— The  pitch  of 
the  rivets  iu  the  following  table  is  for  sheet  steel  of  60,000  Ibs. 
per  square  inch  tensile  strength,  and  for  Swede  Iron  rivets 
of  45,000  Ibs.  per  square  inch  shearing  strength.  The  lap  for 
any  class  or  strength  of  plate  in  the  straight  seams  should 
equal  5  diameters  of  the  rivet  hole  in  single  riveted  joints, 
aid  8  diameters  of  the  rivet  hole  in  double  riveted  joints. 
This  gives  two  diameters  distance  between  edge  of  rivet  hole 
and  edge  of  plate  in  both  single  and  double  riv- 
eted joints,  and  in  double  riveted  joints  also  gives 
two  diameters  (straight  distance)  between  the  two  rows 
of  li vets,  or  three  diameters  straight  across  from  one  pitch 
line  to  the  other.  Such  lap  gives  more  friction  between  the 
plates,  is  more  rigid  and  less  straining  on  the  rivets,  and  may 
be  scarped  down  better  than  a  smaller  lap.  The  round  seams 
should  have  a  lap  of  three  times  the  diameter  of  the  rivet 
ho!e,  and  pitch  as  for  single  riveted  joint. 


SULLIVAN'S  NEW  HYDRAULICS. 
TABLE  No.  39. 


205 


- 

. 

8 

"o 

"S 

| 

5 

r 

|o 

« 

.2 
"3 

1 

S 
o> 
a 

2 

JL 

<D 

g 

S£ 

«D 

ja 

1 

"o  ® 

*o,2 

"S-2 

o 

•S-s 

2xl 

. 

is 

is 

35 

si 

Q 

.2+J 

II 

1 

i 

HW 

•    W  . 

G.No. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

In. 

5. 

0.220 

7-16 

.46875 

.21972656 

1.06875 

1.66725 

2.11-32 

3.34 

6. 

.203 

3-8 

.40625 

lt)r>03906 

0.89405 

1.38185 

2.1-32 

3.14 

7. 

.180 

3-8 

.40625 

.1650:i90t5 

0.95638 

1.50651 

2.1-32 

3.14 

8. 

.165 

3-8 

.40325 

.16503906 

1.00000 

1.64285 

2.1-32 

3.14 

9. 

.148 

14 

.28125 

.07910256 

0.6019r> 

0.92262 

.13-32 

2.1-4 

10- 

.134 

1-4 

.2^12') 

.07910256 

0.63545 

0.98963 

.13-32 

2.1-4 

11. 

.120 

1-4 

.28125 

.07910256 

0.67676 

1.07227 

.13-32 

2.114 

12. 

.109 

3-16 

.21875 

.04785156 

0.48215 

0.74555 

.3-32 

.3-4 

13. 

.095 

3-16 

•21871 

.04785156 

0.52097 

0.82319 

.332 

.34 

14. 

.083 

3-16 

.21875 

.04785156 

0.5646} 

0.91058 

.3-32 

.34 

15. 

.072 

1-8 

.1562^ 

.02441400 

0.35970 

O.F6315 

0.25-32 

.14 

16. 

.065 

18 

.15625 

.02*41406 

0.38161 

0.60697 

0.25-32 

.14 

77— Table  of  Decimal  Equivalents  to  Fractional  Parts 
of  an  Inch. 

The  following  table  will  greatly  facilitate    calculations  of 
of  riveted  joints. 

TABLE  No.  40. 


1-32=.  03125 
1-16=.  0625 
3-32=.  09375 
1-8=.  125 
1-8  +1-32=.  15625 
1-8+116=.  1875 
1-8  +3-32=.  21875 
1-4=  25 
14+1-32=.  28125 
14+1  16=.  3125 

1-4+3-32=.  34375 
3-8=.  375 
3-8+1  32=.  40625 
3-8  +  1-16=.  4375 
38+332=46875 
1-2=.  50 
1-2  +1-32=.  53125 
1-2  +1-16=.  5625 
1-2+3:52=.  59375 
5-8=.  625 

5-8  +1-32=.  65625 
5-8+1-16---.  6875 
58+3-»2=.71875 
34=.  75 
3-4  +1-32=.  78125 
34+1-16=.  8125 
34  +3-32=.  84375 
7-8=.  875 
7-8  +1-32=.  90625 
7-8  +1-16=.  9375 
7-8+3-82=.  96875 

SULLIVAN'S  NEW  HYDRAULICS. 


78.— Weight  of  Each  Thickness,   Per  Square  Foot,   of 
Sheet  Iron  and  Steel 

TABLE  No.  40  A . 


CT1    0 

02    £ 


Thickness 
Inches 


02 


0.300 
.284 
.259 
.238 
.220 
.203 
.180 
.165 
.148 


. 

11.48  lb.- 
10.  47  Ibs 
9.6191bt 

8.8921b< 
8.2051bt 
7.2751bt 


5.9811bs 


11.591bt 
10.571b' 
9.7151b^ 

8.981  ]b 
8.287)bt 
7.  348  Ibs 
6.736  Ibf 
6.041  Jhi- 


0.134 
.120 


.083 
.072 
.065 
.058 
.049 


4.8501bs 
4.  405  Ibs 
3  840  Ibs 
3.  3551  be 
2.9!01bs 
2.  627  Ibs 
2.  344  Ibs 
l 


5.  470  Ibs 
4.  899  Ibs 
4.  449  Ibs 
3.  878  Ibs 
3.3881b8 
2.  939  Ibs 
2.  653  Ibs 
2.  367  Ibs 
1. 


79.- Calculating  Weight  of  Lap- Joint  Riveted   Pipe.— 

In  measuring  the  length  of  a  sheet  of  metal  to  make  a  circle 
of  given  inside  diameter,  allowance  must  be  made  for  the 
contraction  or  compression  of  the  metal  in  bending,  This 
contraction  or  shortening  of  the  plate  in  bending  equals  the 
thickness  of  the  plate  to  be  bent.  Consequently  the  length 
of  plate  required  to  make  a  lap  riveted  pipe  of  a  given  inside 
diameter  in  inches  must  be  equal  to  (d+t)X3.14164-  required 
lap  In  inches.  d=  required  inside  diameter  in  inches,  and 
t=  thickness  of  plate  to  be  bent,  in  inches.  The  weight  of 
the  metal  punched  or  drilled  out  in  making  the  rivet 
holes  for  straight  and  round  seams  is  about  equal  to  25  per 
cent  of  the  weight  of  the  rivets.  Consequently  take  the 
weight  of  the  solid  plate  of  required  dimensions  (Table  No. 
40A)  and  add  75  per  cent  of  total  weight  of  rivets  required. 
If  the  pipe  is  to  be  coated  or  flanged,  this1  must  also  be  added 
to  the  weight.  Allow  for  lap  of  each  round  seam  as  much 
loss  of  length  of  pipe  off  each  sheet  of  metal  as  six  times  the 
diameter  in  inches  of  the  rivet  hole,  except  for  the  two 
sheets  forming  the  ends  of  a  length  of  pipe  which  will  be 

For  the  straight  seams  the  lap  should  be 
Lap=dX8.  for  double  riveted  pipe  joints. 


SULLIVAN'S  NEW  HYDRAULICS.  207 

Lap=dx5,  for  single  riveted  pipe  joints. 

And  for  round  seams  dx3=lap  at  each  end  of  each 
sheet. 

Observe  that  d=  diameter  of  rivet  hole  in  calculating 
lap,  and  in  calculating  the  pitch  of  the  rivets. 

80.—  Tests  for  the  Strength  of  a  Riveted  Lap  Joint- 

To  ascertain  the  actual  net  strength  of  a  riveted  lap  joint  pro- 
ceed as  follows: 

Let  S^tensile  strength  per  square  inch  of  plate  before 
punched. 

S'—  tensile  strength  of  plate  per  square  inch  after 
punched. 

t=thickness  of  plate  in  inches  or  decimals  of  an  inch. 

d=diameter  in  inches  of  rivet  hole. 

P=pitch,  or  distance  from  center  to  center  of  rivets  in 
one  row. 

Then  the  net  tensile  strength  of   the  punched  plate  will 


EXAMPLE. 

The  original  unpunched  plate  had  a  tensile  strength  of 
say  60,000  Ibs.  per  square  inch,  or  S=60,000. 

The  plate  was  of  No.  llguage  steel  and  .12  inch  thick, 
or  t=.12. 

The  diameter  of  rivet  hole  was  d=.28125. 

It  was  double  riveted  and  the  pitch  of  the  rivets  in  one 
row  was  P=1.07227. 

Then, 


The  strength  before  the  rivet  holes  were  made  was 
S=SSXt-60,000=.12X7,200  Ibs. 

Then  the  actual  value  of  S  to  be  used  in  the  formula  for 
thickness  and  strength  of  pipe  shell  (105)  would    be 

=-7377  Per  cent  of  S>  or  60,OOOX-7377. 


208  SULLIVAN'S  NEW  HYDRAULICS. 

The  test  for  actual  strength  of  plate  between  rivet  holes 
in  one  row  being  satisfactory,  we  then  test  the  joint  for  its 
resistance  to  shearing  of  rivets.  The  area  of  net  plate  be- 
tween two  holes  in  one  row  was 

f  P— d  1 
Net  plate  =   [-p  —  J  Xt. 

But  as  the  shearing  resistance  per  square  inch  of  rivet 
metal  was  only  75  per  cent  of  the  tensile  strength  of  the  plate 

60,000 
metal,  we  made  the  rivet  area  =  ^  QQQ=I  33    times    the 

net  plate  area. 

Then,  if  R=  resistance  to  shear  of  rivets,  we  have 

R=  [— p— J  XtXl-333X45,000=5311.48  Ibs. 

This  shows  the  tensile  and  shearing  strength  to  be  equal. 
As  to  test  for  "crippling  strength  of  joint,  we  have  Traut- 
wine's  rule.  N=number  of  rivets  in  one  inch  length  of  joint. 

1  2 

N  —  -p-  for   single  riveted   joint,     and    N=-p-  for  dou- 

joint.      In    this     double     riveted    joint    N= 
=1.86522. 

Then, 

Crippling  strength=NX2  tXdX60,000=7,554  Ibs. 

The  value  of  S  to  be  used  in  formula  (105)  should  be  the 
smallest  of  the  three  values  above  found  if  the  pipe  is  to  be 
absolutely  water  tight,  which  in  this  case  was  S=60,000  X 
.7377  per  cent. 

81.— Testing  Plates  for  Internal  Defects.— The  quality 
of  iron  or  steel  as  to  density  will  of  courpe  be  determined  by 
the  weight  per  cubic  unit  of  the  metal.  Light  weight  indi- 
cates weakness  and  impurities  in  the  metal.  Internal  lamin- 
ations may  be  detected  by  standing  the  plate  on  edge  and 
tapping  it  all  over  with  a  light  hammer.  If  the  sound  is  dull. 


SULLIVAN'S  NEW  HYDRAULICS.  209 

the  plate  is  laminated  internally,  but  if  the  ring  IB  clear  and 
sharp  the  plate  is  sound.  Another  test  is  to  place  supports 
under  the  four  corners  of  the  plate  and  throw  a  thin  layer 
of  dry  fine  sand  upon  the  plate,  and  tap  it  lightly  with  a 
hammer.  If  the  plate  is  defective,  the  sand  will  collect  over 
the  defective  places,  but  if  the  plate  IB  sound  the  vibrations 
will  throw  the  sand  off  the  plate. 

82.— Different  Methods  of  Joining  Pipe  Lengths.— 
Cast  iron  pipe  is  usually  made  in  lengths  of  12  feet,  having 
an  enlargement  at  one  end  of  each  length  called  a  bell  or  hub, 
to  receive  the  spigot  end  of  the  next  length.  After  the  spigot 
is  inserted  into  the  bell  and  adjusted  so  as  to  fit  up  closely  at 
the  end  and  bring  the  pipe  into  line,  a  piece  of  jute,  old  rope, 
or  gasket  cut  long  enough  to  reach  around  the  pipe  with  a 
small  lap,  is  forced  into  the  joint  to  prevent  the  melted  lead 
from  running  into  the  pipe.  A  fire-clay  roll  with  a  rope 
centar  is  now  wrapped  around  the  pipe  cloae  to  the  bell  with 
its  two  ends  turned  out  along  the  top  of  the  pipe  to  guide  the 
melted  lead  into  the  joint.  The  lead  is  made  sufficiently  hot 
to  flow  freely,  and  is  poured  in  until  the  joint  is  full.  The 
lead  is  then  calked  back  into  the  joint  all  around  the  pipe 
with  a  calking  tool. 

Lap  welded  pipe,  such  as  the  converse  lock  joint  pipe 
have  hubs  similar  to  cast  iron  pipe,  and  the  lead  is  poured  by 
the  use  of  a  pouring  clamp.  Lap  welded  and  riveted  pipe  are 
sometimes  joined  by  a  butt  sleeve  joint,  In  this  case  the 
ends  of  two  pipe  lengths  are  butted  evenly  against  each  other 
and  an  iron  or  steel  sleeve  somewhat  thicker  than  the  pipe 
shell,  is  drawn  over  the  joint,  leaving  a  epace  of  %  inch  be- 
tween the  sleeve  and  pipe.  A  little  packing  is  then  inserted 
to  prevent  the  lead  from  running  into  the  pipe,  and  the  space 
between  the  sleeve  and  pipe  is  then  run  full  of  melted  lead. 

When  there  is  too  much  water  in  the  trench  to  permit  of 
pouring  hot  lead  in  pipe  joints,  several  pipe  lengths  may  be 
joined  together  on  the  surface  and  afterward  lowered  into 
the  trench  by  the  use  of  several  derricks,  and  these  compound 
lengths  may  be  jointed  in  the  trench  by  forcing  small  lead 


210  SULLIVAN'S  NEW  HYDRAULICS. 

pipe  into  the  joint  and  setting  it  up  firmly  with  a  calking  tool. 
The  method  of  making  a  slip  joint  was  described  in  §K7. 

83.— Reducers  for  Joining  Pipe  Lengths  of  Different 
Diameters. 

Where  a  pipe  line  is  made  up  of  different  diameters,  or 
where  a  small  pipe  is  to  be  connected  to  a  larger  pipe,  a  re- 
ducer should  be  used  which  is  simply  a  short  length  of  pipe 
converging  from  the  larger  to  the  smaller  diameter.  In  the 
investigation  of  friction  in  nozzles  and  converging  pipes  it 
was  shown  that  the  friction  in  a  converging  pipe  is  much 
greater  than  in  a  uniform  pipe  whose  diameter  is  equal  to 
the  mean  or  average  diameter  of  the  converging  pipe.  The 
friction  in  a  converging  pipe  depends  upon  its  length  and 
mean  diameter.  Its  mean  diameter  should  be  as  great  as 
possible  and  its  length  as  short  as  possible  provided  it  does 
not  converge  more  rapidly  or  at  a  greater  angle  than  the  form 
of  the  vena  contracta  or  contracted  vein.  If  d  is  the  inside 
diameter  of  the  larger  pipe,  then  in  a  length  of  the  reducer 

equal-o~,  the  diameter  should  converge  to  d'=dX-7854.  For 
example  a  pipe  of  20  inches  diameter  is  to  be  joined  to  a  pips 
of  3  inches  diameter,  and  it.  is  required  to  find  the  length  of 
the  reducer  in  inches. 

Let  d  =  diameter  in  inches  of  the  large  pipe=20. 

d'=diameter  in  inches  of  the  small  pipe=3. 
d         20 

Then,  in  a  length=-^-  =~2~  =  1°    inches,    the  reducer 

must  converge  to  a  diameter=dX.7854  =  20  X  -7854  =  15.708 
inches.  Hence  total  amount  of  convergence  is  d — d'=20 — 
15.708=4.295  inches  in  a  length  of  10  inches,  or  the  rate  of 

convergence  per  inch  length  of  the  reducer  is—  ,Q*J  =.4292  of 
an  inch  per  inch  length.  Or  the  diameter  will  converge  1 
inch  in  a  length  of  ^92  =2-33  inches.  It  shoutd  there- 
fore converge  from  20  to  3  inches  diameter  in  a  length  I  = 
(d— d')X2.33=(20— 3)X2.33=39.61  inches.  If  a  diameter  of 


SULLIVAN'S  NEW  HYDRAULICS.  211 

one  foot  is  to  be  joined  to  a  diameter  of  .7854  foot,  then  the 
length  in  feet  of  the  reducer  should  be  I  s=  (d— d')  X  2.33= 
(1.— .7854)X2.33=  .5  foot. 

In  this  latter  case  d  and  d'  are  expressed  in  feet.  All 
reducers  and  all  nozzles  for  fire  streams  or  power  mains,  and 
conical  pipes  in  general  should  conform  to  the  foregoing  pro- 
portions where  the  most  effective  delivery  and  smallest  loss 
by  friction  and  contraction  are  desired. 

The  rate  of  convergence  is  one  inch  in  2.33  inches  length 
or  one  foot  in  2.33  feet  length  of  the  converging  pipe  and 
hence  the  length  of  the  convergent  pipe  or  reducer  will  be 
found  by  the  general  formula 

f=(d— d')X2.33 (107) 

If  I  is  expressed  in  inches  then  d  and  d*  must  be  in  inches 

If  l=is  in  feet,  then  d  and  d'  must  be  in  feet. 

d=largest  diameter. 

d'=smallest  diameter. 


CHAPTER  VI. 


Plow  in  Open  Channels  of  Uniform  Cross  Section. 


84. — Permanent  and  Uniform  Flow. — Permanent  flow 
may  .occur  in  a  channel  either  of  uniform  or  non-uniform 
cross  section.  The  flow  is  said  to  be  permanent  when  an 
equal  quantity  flows  through  each  cross  section  in  equal 
timea.  If  the  cross  sections  of  the  channel  are  of  unequal 
area  the  velocities  will  be  inversely  as  the  areas,  in  the  case 
of  permanent  flow.  Uniform  flow  can  only  occur  in  a  chan- 
nel of  uniform  cross-section  and  grade.  By  uniform  flow  is 
meant  that  both  the  mean  velocity  and  the  quantity  are 
equal  at  all  places  along  the  channel.  In  this  case  the  slope 
of  the  water  surface  and  the  slope  of  the  bottom  of  the  chan- 
nel are  necessarily  the  same,  otherwise  the  velocities  or  quan- 
tities passing  different  points  would  not  be  equal.  In  natural 
streams  with  firm  beds  which  are  not  undergoing  scour  and 
fill,  the  flow  will  become  permanent  if  the  supply  of  water  is 
constant  and  uniform.  These  conditions  can  scarcely  occur 
in  large  streams  of  great  length,  but  may  occur  in  email  riv- 
ers or  creeks.  In  artificial  channels  such  as  irrigation  canals 
and  mill  races  where  the  area  of  cross  section  and  grade  are 
uniform,  and  where  the  quantify  admitted  into  the  canal  is 
constant  and  uniform,  both  permanent  and  uniform  flow  will 
occur  after  sufficient  time  has  elapsed  for  equilibrium  to  be 
established  between  the  acceleration  of  gravity  and  the  re- 
sistances to  flow,  provided  seepage  and  evaporation  are  not 
appreciably  great,  as  sometimes  they  are. 

If  different  portions  of  a  canal  are  all  of  uniform  sectional 
area  but  the  slope  is  different  in  the  different  divisions,  the 
flow  may  become  permanent,  but  cannot  become  uniform  un- 
less the  roughness  and  resistances  in  the  portions  of  greatest 
slope  happen  to  be  just  enough  greater  than  in  the  other  divis- 
ions to  equalize  the  velocity  head  in  all.  In  such 


SULLIVAN'S  NEW  HYDRAULICS.  213 

case  each  division  might  be  considered  separately 
and  the  flow  might  be  called  uniform  in  and  for 
any  given  division  of  the  canal  in  which  the 
area  and  slope  are  uniform.  With  the  exception  of  flumes 
aqueducts,  and  canals  lined  with  masonry,  there  are  few  open 
channels  in  which  uniform  flow  takes  place.  The  variations 
in  grade,  area  of  cross-section  and  roughness  of  perimeter 
may  each  be  slight  and  yet  the  effect  is  marked.  In  uni- 
form flow  the  resistances  and  accelerations  of  gravity  must 
be  constantly  equal  to  each  other.  If  the  slope  varies  the 
head  will  be  greater  in  one  division  than  in  another.  If  the 
sectional  area  varies  the  resistances  will  be  inversely  as  y'r8, 
and  will  also  be  increased  by  cross  currents  and  re  actions  of 
the  particles  of  water  which  impinge  upon  the  irregularities 
of  the  perimeter  and  react  therefrom.  The  resistances  due 
to  mere  irregularities  of  perimeter  are  similar  to  the  resist- 
ances of  a  bend  in  a  pipe  or  open  chancel  .  They  deflect  the 
particles  of  water  impinging  upon  them  and  thus  destroy  an 
amount  of  head  depending  upon  the  angle  of  deflection  and 
the  velocity  of  the  particles  affected. 

Where  the  width  of  a  channel  is  alternately  small  and 
then  greater,  the  resistances  are  similar  to  these  in  a  con- 
vergent or  divergent  pipe,  and  will  vary  with  the  mean  value 
of  i/r3  for  a  given  convergent  length  of  channel  and  with  the 
mean  velocity  through  the  section  having  the  mean  value  of 
r  for  the  given  length  considered.  It  is  apparent,  therefore, 
that  a  coefficient  which  would  apply  at  one  station  or  to  one 
given  short  length  of  a  non-uniform  channel,  will  not  apply 
at  another  station  or  to  another  given  length  unless  the  same 
conditions  of  roughness  and  convergence  of  banks  obtain  at 
both. 

In  natural  streams  containing  bende  of  varying  grade, 
depth  and  width,  there  will  be  what  may  be  termed  velocity 
of  approach  in  many  of  its  divisions  which  will  cause  veloci- 
ties in  short  straight  reaches  of  the  channel  which  are  ap- 
parently greater  than  the  velocity  due  to  the  apparent  slope. 
A  coefficient  of  velocity  C,  developed  from  the  data  of  flow 


214  SULLIVAN'S  NEW  HYDRAULICS. 

observed  at  such  places  will  be  much  too  high  to  be  appli- 
cable at  any  other  reach  or  to  any  other  conditions  of  flow. 
Such  conditions  are  most  common  at  low  water  stages,  and 
may  not  obtain  at  the  same  place  during  medium  and  high 
stages  of  water.  In  natural  and  non-uniform  channels  the 
areas  for  different  depths  of  flow  and  the  various  angles 
made  by  the  banks  at  different  heights,  and  the  varying  de- 
grees of  roughness  of  the  banks  above  the  usual  depth  of 
flow,  so  complicate  the  conditions  for  different  depths  of  flow 
at  any  given  station  that  it  is  necessary  to  find  the  value  of  C 
for  the  given  station  under  each  separate  set  of  conditions. 
In  Section  13  an  approximate  method  of  determining  C  under 
such  conditions  has  been  pointed  out.  It  will  require  a  con- 
sideration of  the  form  of  the  channel  above  and  below  the 
observation  station  as  well  as  at  the  station.  No  one  formula 
without  the  aid  of  auxiliary  formulas,  such  as  suggested 
in  §  13,  supplemented  by  experience  and  sound  judgment, 
can  be  made  to  apply  to  the  conditions  of  flow  in  rivers  and 
irregular  channels.  With  all  attainable  aids,  we  can  only 
expect  fairly  approximate  results  in  such  cases.  We  shall 
therefore  consider  the  flow  in  artificial  channels  of  uuiform 
grade  and  sectional  area,  or  channels  in  which,  by  courtesy, 
these  conditions  are  said  to  be  approximated.  It  would  be 
closer  the  truth  to  say  that  the  flow  is  permanent  to  a  degree 
approaching  uniform  flow  in  each  division  of  uniform  slope. 

The  closer  the  actual  conditions  approach  to  uniform 
flow  the  closer  will  be  the  computed  results  by  the  formula 
for  flow. 

85.—  Resistances  and  Net  Mean  Head  In  Open  Chan- 
nels.— In  channels  of  uniform  grade  and  cross  section  the  re- 
sistances to  flow  consist  in  the  friction  of  the  liquid  in  contact 
with  the  perimeter  and  the  internal  resistances  among  the 
particles  of  water  themselves.  The  internal  resistances  are 
caused  by  the  distortion  of  the  onward  course  of  some  of  the 
particles  of  water  causing  them  to  collide  with  and  distort  the 
course  of  other  particles. 

These  distortions  have  their  origin  in  the  small   inoquali- 


SULLIVAN'S  NEW  HYDRAULICS.  215 

ties  or  roughnesses  along  the  sides  and  bottom  of  the  channel 
against  which  the  moving  particles  flow,  and  from  which 
they  are  hurled  off  in  eddies  angling  across  the  path  of  the 
parallel  flow.  Difference  in  the  temperature  of  different  par- 
ticles of  water,  which  may  be  caused  in  part  by  impact  and 
velocity,  also  causes  upward  and  downward  movements 
among  the  particles  of  water.  If  each  particle  of  water 
moved  uniformly  in  a  course  parallel  to  the  bottom  and  sides 
the  term  resistance  to  flow  would  include  no  element  of  any 
importance  except  what  is  called  the  friction  of  the  liquid 
with  the  solid  perimeter.  The  results  of  experiments  estab- 
lish the  fact,  however,  that  the  sum  total  of  all  the  resist- 
ances whether  internal  or  of  friction  at  the  perimeter,  are 
proportional  to  the  extent  of  wetted  perimeter,  in  channels  of 
uniform  cross  section  and  slope.  The  internal  resistances 
among  the  particles  of  water  are  not  caused  by  friction  of  one 
particle  with  another,  but  by  the  collisions  and  reactions  of 
particles  travelling  in  different  directions.  There  can  be  no 
friction  as  between  the  particles  themselves  for  they  have  no 
roughnesses  to  interlock  or  by  which  they  can  take  hold  on 
each  other. 

The  molecules  are  independent,  free  bodies   which  act 
upon  each  other  by  impact  only,  and  not  by  friction. 

If  the  flow  could  occur  without  any  resistance  of  any 
nature  the  effect  of  gravity  would  accelerate  the  flow  so  that 
the  rate  of  velocity  at  any  given  point  down  a  uniform  grade 
would  equal  the  square  root  of  the  total  fall  in  feet  between 
the  origin  of  flow  and  *he  given  point.  The  velocity  on  a  uni- 
form grade  would  therefore  constantly  increase  each  second. 
As  this  result  does  not  actually  occur,  but  on  the  contrary 
the  mean  velocity  becomes  uniform  throughout  the  length 
of  such  grade,  it  is  evident  that  the  acceleration  of  gravity 
has  been  balanced  by  and  is  equal  to  the  combined  resist 
ances  to  flow.  It  is  equally  evident  that  the  resistances  are 
as  the  square  of  the  velocity  or  are  equal  to  the  total  head  in 
each  foot  length.  If  this  were  not  true  there  would  be  a  gain 
in  unresisted  head  in  each  foot  length  of  channel,  and  to  this 
extent  the  acceleration  of  gravity  would  cause  the  velocity  to 


216  SULLIVAN'S  NEW  HYDRAULICS. 

increase  in  each  foot  length  of  channel,  and  there  could  be  no 
such  thing  as  uniform  flow  under  any  conditions,  and  all  for- 
mulas based  upon  the  theory  of  uniform  flow  would  necessar- 
ily fail  .Attention  was  called  to  this  in  the  discussion  of  coeffi- 
cients and  the  law  of  variation  of  coefficients.  It  is  mentioned 
again  here  because  some  hydraalicians  of  eminent  ability 
contend  that  the  coefficient  of  friction  or  rather  of  resistance 
will  decrease  with  an  increase  in  the  velocity,  which  means 
that  the  acceleration  of  gravity  is  greater  than  the  combined 
resistances  to  flow.  If  that  contention  can  be  established  it 
must  be  admitted  that  uniform  flow  is  an  impossibility  either 
in  pipes  or  in  channels  of  uniform  grade  and  cross  section. 
The  writer  is  not  yet  ready  to  make  that  admission.  The  law 
governing  the  flow  in  pipes  of  uniform  diameter  is  the  same 
which  governs  the  flow  in  all  uniform  channels.  The  theory 
of  flow  and  resistance  to  flow  was  discussed  in  general  hereto- 
fore (§3  to  7  inclusive)  and  need  not  be  repeated  here. 

It  is  evident  that  the  velocity  of  any  given  film  or  parti- 
cle of  water  will  depend  upon  the  net  unresisted  head  of  such 
film  or  particle  after  the  resistances  to  its  flow  have  been 
balanced.  It  is  equally  evident  that  the  mean  of  all  the  dif- 
ferent velocities  in  a  cross  section  will  depend  upon  the 
mean  net  head  of  all  the  particles.  If  the  mean  net  head  in- 
creases more  rapidly  than  the  resistances,  it  follows  that  the 
rate  of  velocity  will  increase  in  every  successive  foot  length 
of  channel;  which  we  know  is  not  the  case.  In  channels  of 
uniform  grade  and  cross  section  the  sum  of  the  resistances 
per  foot  length  of  channel  is  equal  to  the  head  included  in 
each  foot  length,  and  tbus  leave  the  net  unresisted  head,  or 
velocity  head,  a  uniform  and  constant  quantity,  and  the  uni- 
form mean  velocity  is  as  the  square  root  of  this  constant  net 
mean  head. 

There  is  no  friction  between  the  molecules  of  the  atmos- 
phere and  the  molecules  of  water  at  the  surface.  Before  fric- 
tion can  occur  between  two  independent  bodies  it  is  neces- 
sary that  both  of  the  bodies  should  have  projections  or  rough- 
nesses which  would  interlock,  and  require  force  to  separate. 


SULLIVAN'S  NEW  HYDRAULICS.  217 

When  winds  occur,  the  molecules  of  air  are  hurled  against 
the  molecules  of  water  and  thus  create  resistance  by  distort- 
ing the  course  of  the  water  from  its  direct  path,  if  the  direc- 
tion of  the  wind  if  not  the  same  as  that  of  the  flow,but  if  the 
wind  follows  the  direction  of  the  flow  of  the  water  with  a 
downward  sweep  it  does  not  resist,  but  assists  the  flow.  The 
small  bombardment  of  the  water  surface  by  molecules  of  air 
caused  by  difference  in  temperature  of  different  air  strata 
does  not  cause  any  appreciable  resistance  to  or  distortion  of 
the  flow.  In  truth,  it  may  be  said  that  none  of  the  resist- 
ances to  flow  are  due  to  pure  friction,  but  are  all  due  to 
changes  in  direction  of  the  courses  of  different  molecules 
which  produces  internal  collisions  and  reactions  as  well  as 
collisions  with  and  reactions  from  the  solid  perimeter.  The 
projections  and  inequalities  along  the  perimeter,  however 
small  they  may  be,  distort  the  course  of  the  molecules  of 
water  impinging  upon  them,  and  the  reaction  sends  them  ed- 
dying across  the  path  of  the  adjacent  molecules  causing  fur- 
ther distortions  and  reactions  among  the  molecules  them- 
selves. Roughnesses  along  the  bottom  of  a  channel  cause 
whirls  and  boils  and  vertical  currents  which  spend  their  en- 
ergies in  reaching  the  water  surface  and  there  spread  out  in- 
ert and  without  direction  or  velocity.  For  this  reason  the  ve- 
locity at  the  surface  is  less  than  it  is  below  the  surface,  which 
fact  has  led  some  persons  to  believe  that  there  is  friction  be- 
tween the  atmosphere  and  the  water  surface. 

Such  boils  rise  above  the  surface  of  the  water  on  the 
same  principle  that  water  rhes  above  the  surf  ace  in  a  Pitot 
tube,  and  when  it  reaches  the  height  due  to  its  velocity,  its 
energy  is  spent,  and  it  spreads  out  in  all  directions  upon  the 
surface.  Abrupt  bends  or  changes  in  the  direction  of  flow 
produce  impact  and  reaction  and  cause  the  formation  of 
whirls  and  cross  currents  which  are  finally  overcome  by  con- 
tact with  the  onward  flow  at  the  expense  of  considerable  head, 
the  amount  of  which  will  depend  upon  the  angle  and  the 
radius  of  the  bend.  These  remarks  in  connection  with  the 
laws  of  resistance  given  at  §  -.  and  the  discussion  of  the  re- 


218  SULLIVAN'S  NEW  HYDRAULICS. 

lationsof  area  to  wetted  perimeter  and  the  resulting  relations 
between  acceleration  and  resistance  discussed  in  §§  3  to  7 
both  inclusive,  it  is  believed  will  cover  all  the  important  fea- 
tures relating  to  flow  and  resistance  to  flow  in  channels  of 
uniform  grade  and  sectional  area.  There  are,  however 
certain  ratios  and  relations  of  surface,  to  mean  and  bottom 
velocities  in  open  channels  which  demand  a  separate  and 
more  special  investigation,  as  the  knowledge  of  these  re- 
lations has  always  been  involved  in  much  uncertainty.  The 
writer's  theory  of  these  relations  is  entirely  original,  and  is 
based  upon  his  theory  of  coefficients  of  resistance  and  upon 
observation  and  experiment. 

86. —There  is  no  Constant  Ratio  Between  the  Surface, 
the  Mean  and  the  Bottom  Velocities. 

It  cannot  be  denied  that  the  velocity  of  flow  of  any  given 
particle  of  water  will  depend  wholly  upon  the  net  unresisted 
head  of  such  particle. 

The  conditions  under  which  the  motion  of  any  given  par- 
ticle takes  place  will  vary  with  the  relative  position  of  the 
particle  in  the  cross  section  with  reference  to  the  perimeter, 
which  is  the  original  place  of  impact  and  reaction.  The  dis- 
tance that  a  rebounding  particle  will  be  projected  into  and 
across  the  flow  will  depend  upon  the  difference  in  the  velocity 
along  and  near  the  perimeter  and  the  velocity  at  the  center 
and  surface  of  the  cross  section,  or  the  difference  in  the 
velocity  of  the  rebounding  particle  and  that  of  the  particles 
with  which  it  comes  in  collision.  The  action  of  a  particle  of 
water  is  similar  to  that  of  a  billiard  ball.  When  it  impinges 
upon  a  projection  along  the  perimeter  its  course  is  changed 
so  that  it  travels  diagonally  toward  the  opposite  bank  or 
surface,  but  instantly  meets  the  opposition  of  the  particles 
having  a  direction  of  flow  parallel  to  the  perimeter. 

The  force  and  direction  of  the  reaction  is  changed  and 
reduced  with  each  successive  collision  as  the  rebounding 
particle  travels  across  the  parallel  flow,  until  its  direction 
also  becomes  parallel  and  the  resistances  and  collision  cease 
as  to  that  particle. 


SULLIVAN'S  NEW  HYDRAULICS  219 

These  impingements  and  reactions  along  the  sides  and 
bottom  are  in  continual  progress  and  are  naturally  stronger 
at  the  place  of  their  origin  along  the  perimeter  than  else- 
where and  grow  weaker  and  weaker  as  they  approach  the 
center  of  the  volume  of  flow.  The  number  of  these  reactions 
will  be  directly  as  the  roughness  of  the  perimeter.  If  the 
bottom  of  the  channel  is  corrugated  transversely  the  entire 
volume  of  water  will  rise  and  fall  and  reproduce  the  corru- 
gations on  the  surface,  thus  agitating  the  entire  volume  of 
flow. 

In  such  case  there  will  be  only  a  small  difference  in  the 
surface  velocity  and  that  at  mid-depth,  but  the  bottom  ve- 
locity will  be  almost  nothing.  If  the  sides  and  bottom  of  the 
channel  are  fairly  uniform  and  smooth  there  will  be  very 
little  disturbance  at  the  surface  and  a  small  number  of  re- 
actions from  the  bottom  and  sides,  and  the  bottom  velocity 
will  be  proportionately  much  greater,  which  will  result  in  in- 
creasing the  mean  velocity.  It  is  well  known  that  the  mean 
velocity  will  increase  very  rapidly  in  uniform  channels  or  di- 
ameters, simply  by  increasing  the  hydraulic  mean  radius 
without  increasing  the  slope.  This  is  accounted  for  by  the 
fact  that  as  diameter  or  hydraulic  mean  radius  increases,  the 
area  of  cross  section  of  the  column  of  water  gains  very  rap- 
idly on  solid  perimeter  aud  there  will  be  a  very  large  rela- 
tive quantity  passed  which,  in  smooth,  uniform  channels  of 
large  radius,  will  not  come  in  contact  with  the  perimeter  nor 
any  other  retarding  influence.  The  result  ie  to  increase  the 
rate  of  mean  velocity,  not  by  increasing  the  bottom  velocity 
but  by  increasing  the  area  or  section  of  the  unretarded  por- 
tion of  the  vein,  or  the  number  of  particles  of  water  having 
an  unresisted  head.  An  increase  in  hydraulic  mean  radius  or 
of  diameter  can  not  affect  the  velocity  of  the  water  in  con- 
tact with  the  perim  iter  or  affected  thereby.  It  do  '8  not  re- 
move the  resistance  nor  add  anything  to  the  net  head  or 
freedom  of  flow  of  these  particles.  An  increase  in  hydraulic 
radius  or  diameter  cannot  relieve  the  roughness  ol  the  peri- 
meter nor  the  reactions  therefrom,  nor  does  it  ad  I  anything 


220  SULLIVAN'S  NEW  HYDRAULICS. 

to  their  head.  There  is  no  conceivable  reaeon.therefore,  why 
the  bottom  velocity  should  increase  or  decrease  with  changes 
in  hydraulic  mean  depth  or  diameter,  because  it  will  be  af- 
fected by  the  same  retarding  influences  and  resistances  re- 
gardless of  the  value  of  the  diameter  or  hydraulic  radius. 
The  velocity  along  the  sides  and  bottom  of  a  channel  will 
therefore  depend  solely  upon  thd  degree  of  roughness  of  the 
wetted  perimeter  and  the  slope  of  the  channel,  and  will  in  no 
manner  be  affected  by  an  increaee  in  the  hydraulic  radius  or 
size  of  the  channel.  It  cannot  be  maintained  that  the  rapid 
movement  of  the  upper  central  core  of  the  liquid  vein  will 
assist  the  flow  at  the  sides  and  bottom,  because  the  minute 
globules  of  water  are  independent  of  each  other  anJ  are 
without  friction  among  themselves.  There  are  no  rough- 
nesses upon  these  globules  of  water  by  which  they  can  take 
the  slightest  hold  on  each  other.  If  there  were  any  rough- 
nesses upon  them  they  would  interlock  and  the  flow  would 
become  uniform  and  as  great  at  the  perimeter  as  at  the  cen- 
ter, or  would  be  brought  to  rest  entirely  by  friction  with  the 
perimeter.  There  is  nothing  to  affect  the  velocity  of  flow 
of  any  particular  portion  of  the  vein  except  the  constant 
net  head  it  has  remaining  after  the  resistances  to  its  flow 
have  been  balanced.  As  an  increase  in  hydraulic  mean  rad- 
ius cannot  relieve  the  roughness  and  reaction  at  the  peri- 
meter and  the  consequent  loss  of  head  to  the  portion  of  the 
vein  thus  affected,  it  cannot  therefore  increase  its  velocity 
which  must  depend  solely  upon  the  inclination  of  the  chan- 
nel and  roughness  of  perimeter  The  velocity  of  the  water 
affected  by  the  perimeter  will  be  the  same  for  the  same  slope 
and  same  degree  of  roughness  regardless  of  the  sizj  of  the 
channel  and  regardless  of  the  mean  and  surface  velocity. 
This  is  directly  confirmed  by  the  fact  that  very  high  mean 
and  surface  velocities  may  be  permitted  in  large  canals  with- 
out damage  by  erosion  of  the  bed,  while  such  mean  velocity 
in  a  small  canal  would  rapidly  destroy  its  bed.  The 
reason  is  that  the  small  canal  would  require  a  steep  slope 
to  generate  a  high  mean  velocity  because  the  whole  volume 


SULLIVAN'S  NEW  HYDRAULICS.  221 

of  water  in  a  small  canal  is  affected  by  the  resistance  of 
and  reactions  from  the  perimeter,  and  consequently  the  bot- 
tom velocity  which  is  controlled  by  the  slope,  would  be  dis- 
astrously high. 

The  smoother  the  perimeter,  the  fewer  the  reactions  and 
disturbances,  and  the  greater  the  area  of  cross  section  un- 
affected by  retarding  influences,  and  as  the  area  of  unresisted 
section  increases,  the  mean  velocity  will  increase.  In  such 
case  the  ratio  of  surface  to  mean  velocity  will  be  small  but 
the  ratio  of  bottom  to  mean  or  surface  velocity  will  be  great. 
The  mean  velocity  is  apparently  largely  controlled  by  the 
ratio  of  area  to  perimeter  as  well  as  by  smoothness  of  peri- 
meter and  slope  of  channel. 

The  bottom  velocity  is  controlled  entirely  by  the  slope  and 
the  roughness  of  perimeter.  After  the  depth  of  flow  is  sufficient 
to  remove  the  water  surface  from  the  small  reaction  from  the 
bottom  in  a  fairly  smooth  channel,  the  surface  velocity  de- 
pends only  upon  the  slope  and  nothing  else. 

It  is  evident  that  there  is  no  fixed  ratio  between  any  two 
of  these  three  velocities.  The  different  velocities  are.'dependi 
ent  upon  separate  and  distinctly  different  conditions.  The 
mean  velocity  gains  as  area  gains  over  perimeter  without  any 
increase  of  slope,  not  because  the  maximum  velocity  gains,  but 
because  a  greater  number  of  particles  are  set  free  from  the 
retarding  influences  of  the  perimeter  and  thus  increase  the 
sectional  area  of  the  vein  having  the  higher  velocity.  This 
does  not  affect  the  bottom  velocity  because  there  is  no 
change  of  slope.  If  the  channel  is  comparatively  deep  and 
has  a  smooth  bottom,  a  further  increase  in  hydraulic  mean 
depth  would  not  affect  the  maximum  surface  velocity  which, 
under  these  circumstances  would  be  removed  from  the 
effects  of  reactions  from  the  bottom  and  would  therefore 
only  be  increased  by  an  increase  of  slope  simply.  It  is  evi- 
dent that  the  relation  of  the  maximum  surface  velocity  to 
the  bottom  velocity  is  more  constant  than  the  relation  of 
surface  to  mean  or  of  mean  to  bottom  velocity,  and  it  is  also 
evident  that  there  are  so  many  different  influences  affecting 
the  one  which  does  not  affect  the  other  to  an  appreciable 
degree,  that  it  cannot  be  said  that  there  is  any  given  ratio  or 
relation  between  any  two  of  them. 


222  SULLIVAN'S  NEW  HYDRAULICS. 

The  relation  between  them  will  be  very  different  in  a 
shallow  rough,  stony  channel  from  what  it  will  be  in  a  deep 
smooth  channel,  and  the  relation  will  change  in  any  given 
channel  with  changes  in  depth  of  flow.  It  has  been  demon- 
strated that  the  mean  velocity  will  increase  as  £/r3  while  all 
other  conditions  remain  constant.  The  increase  in  r  does 
not  affect  the  bottom  velocity  at  all.  An  increase  in  r  may  or 
may  not  increase  the  maximum  surface  velocity.  The 
various  empirical  formulas  for  deducing  the  mean  or  the 
bottom  velocity  from  the  surface  velocity  are  therefore 
totally  unreliable,  for  such  a  formula  can  only  apply  to  one 
set  of  given  conditions.  If  such  formula  would  apply  to  a 
wooden  trough  two  feet  wide  and  one  foot  deep,  it  would  not 
apply  to  a  canal  five  feet  wide  and  three  feet  deep.  If  it 
would  apply  to  a  canal  with  smooth  and  uniform  perimeter 
it  would  not  apply  to  a  rough  canal  of  like  dimensions.  Such 
formulas  are  therefore  not  of  sufficient  importance  to  de- 
mand discussion. 

87. — The  Eroding  Velocity  in  Unpaved  Channels  in 
Earth. 

In  irrigation  engineering  there  is  no  one  feature  of  greater 
importance  than  the  proper  adjustment  of  the  eroding 
velocity,  or  velocity  adjacent  to  the  sides  and  bottom,  to  the 
character  of  the  soil  which  must  form  the  perimeter  of  the 
canal.  There  is  one  particular  bed  velocity  best  adapted  to 
each  different  class  of  earth.  From  considerations  of  econ- 
omy it  is  desirable  to  maintain  as  high  a  velocity  as  the 
nature  of  the  material  forming  the  canal  bed  will  stand 
without  damage  by  erosion. 

The  stability  of  the  bed  of  a  canal  will  depend  upon  (1) 
the  nature  of  the  material  forming  the  bed,  (2)  the  alignment 
of  the  canal.  (3)  the  angle  made  by  the  side  slopes,  (4)  the  vel- 
ocity of  flow  of  that  portion  of  the  vein  adjacent  to  the  sides 
and  bottom,  (5)  the  action  of  frost,  or  climatic  influences. 

The  destruction  of  the  side  slopes  depends  as  much  or 
more  upon  the  angle  made  by  them  as  upon  the  velocity  of 
flow  in  contact  with  and  adjacent  to  them.  In  cold  climates 


SULLIVAN'S  NEW  HYDRAULICS.  223 

where  frost  penetrates  the  earth  to  a  depth  of  several  feet 
the  side  slopes  should  be  much  flatter  for  the  same  nature  of 
material  than  in  climates  not  subject  to  frost. 

The  eroding  velocity  in  a  majority  of  cases  is  only  the 
partial  agent  of  destruction  of  the  bed.  Bad  alignment  and 
side  slopes  too  steep  to  withstand  the  disintegrating  action 
of  alternate  freezings  and  thawings  are  the  principal  factors 
in  destroying  the  uniformity  and  efficiency  of  the  canal. 

In  a  canal  of  uniform  section  with  direct  alignment  the 
only  velocity  which  tends  to  erode  the  perimeter  is  the  vel- 
ocity of  the  water  which  is  in  contact  with  it,  which  velocity 
is  governed  entirely  by  the  slope  and  roughness  of  peri- 
meter and  is  not  affected  by  the  value  of  the  hydraulic  mean 
depth. 

On  the  contrary  if  the  canal  has  bends  and  curves,  then 
the  surface,  mean  and  bottom,  and  all  intermediate  vel- 
ocities, become  eroding  velocities  at  all  places  where  the 
direction  of  flow  is  changed.  The  outer  bank  of  the  curve 
must  form  the  resistance  which  forces  the  change  in  direc- 
tion of  flow.  The  amount  of  this  resistance  will  depend  upon 
the  amount  of  change  in  direction  of  flow  and  the  time  or 
distance  in  which  the  change  is  finally  effected.  It  requires 
work  and  power,  (see  §60)  The  resistance  will  therefore  be 
distributed  along  the  outer  curves  over  a  distance  depending 
upon  the  abruptness  of  the  curve  or  upon  the  distance  in 
which  the  total  curvature  is  effected.  The  power  expended 
upon  each  square  unit  of  area  of  the  outer  curve  will  there- 
fore be  directly  as  the  radius  of  the  curve.  This  is  the 
measure  of  resistance  which  each  unit  of  area  must  be 
sufficiently  stable  to  offer,  otherwise  it  will  be  eroded  and 
removed. 

A  comparison  of  the  coefficients  for  straight  flumes  with 
the  coefficient  of  the  crooked  Highlme  flume  (Group  No.  5) 
would  indicate  that  the  resistance  of  a  bend  of  90°  with  a 
radius  equal  one-half  the  width  of  the  channel  would  amount 
to  at  least  twice  the  head  iu  feet  generating  the  mean  velocity 
of  flow.  If  this  ratio  of  resistance  holds  good  in  channels  of 


224  SULLIVAN'S  NEW  HYDRAULICS 

all  widths  then  the  resistance  (which    is   equal   to   the   head 
required  to  balance  it)  would  be 

A        2v*X.  007764       AX2v2X.OQ776* 

^go^X      ~~R~~  9oxR       (108) 

In  which 

A=angle  in  degrees  included  in  central    arc  of  bend. 
R=radius  of  central  arc  of  bend   in   widths  of   the 
channel,  not  feet. 

For  further  discussion  see  §  69  et  seq.,  where  the  for- 
mula is  explained  in  detail. 

In  channels  with  converging  banks  the  resistance,  which 
they  must  be  sufficiently  stable  to  offer  and  withstand  is 
similar  to  that  in  a  conical  or  convergent  pipe(§§  37,39),  and 
therefore  will  vary  as  (3Xv)s,  when  v=  the  mean  velocity 
through  the  section  of  the  convergent  length  at  the  point 
where  the  value  of  r  is  the  mean  or  average  value  of  r  for  the 
whole  length  of  the  convergent  channel.  If  the  channel  is 
both  curved  and  convergent  at  the  same  place,  then  the 
banks  must  be  able  to  withstand  the  resistances  due  to  both 
causes.  The  necessity  of  direct  alignment  and  of  uniformity 
of  cross-section  is  therefore  apparent,  if  we  would  avoid 
erosion  and  yet  maintain  a  reasonably  high  mean  velocity. 
In  large  rivers  which  have  small  slope  and  frequent  bends 
with  cross-sections  alternately  wide  and  shallow  and  then 
deep  and  narrow,  all  the  velocities  become  eroding  velocities 
and  their  forces  vary  inversely  as  !/r8.  The  work  done  by 
the  impinging  water  is  in  the  direction  of  straightening  the 
bends  and  trimming  the  sides  so  the  width  will  be  uniform, 
and  in  bringing  the  slope  of  the  bottom  to  uniform  grade. 
Unfortunately  the  banks  and  bends  cave  in  and  form  new 
resistances  which  divert  the  energies  and  directions  of  the 
water  to  new  quarters,  and  thus  its  work  is  self  destructive. 
In  artifical  channels  this  work  should  be  done  in  advance  so 
that  the  energies  of  the  water  may  be  employed  in  a  profit- 
able way,  and  not  wasted  in  building  and  destroying  bars  and 
bends. 


SULLIVAN'S  NEW  HYDRAULICS.  225 

88.—  Eroding  Velocity  in  Straight  Canals  of   Uniform 

Section.—  Theory  and  observation  both  indicate  that  a  depth 
of  flow  of  one  foot  upon  the  perimeter  of  a  straight  canal  of 
uniform  section  will  cause  as  great  erosion  as  a  flow  of  ten 
feet  depth  or  any  greater  depth.  The  power  of  erosion  in  a 
straight,  uniform  canal  varies  with  the  square  of  the  bottom 
velocity,  or  as  the  square  of  the  velocity  in  contact  with  the 
sides  and  bottom.  It  has  been  shown  that  the  velocity  along 
the  sides  and  bottom  is  controlled  by  the  slope  and  degree  of 
roughness  of  perimeter,  in  straight  uniform  channels,  and  that 
this  velocity  cannot  be  affected  in  such  channels  by  any 
change  in  hydraulic  mean  radius. 

As  this  bed  velocity  is  not  affected  by  the  size  of  the  chan- 
nel, but  is  the  same  for  the  same  slope  of  channel  bed  and 
roughness  of  perimeter  without  regard  to  hydraulic  mean 
radius  of  the  channel,  we  may  conceive,  for  the  purpose  of  de- 
termining the  eroding  velocity  in  such  straight  uniform  chan- 
nel, that  the  central  portion  of  the  liquid  vein  has  been  re- 
moved so  that  there  remains  only  one  foot  depth  of  water 
upon  the  sides  and  bottom  of  the  channel. 

Then  find  the  sectional  area  of  this  layer  of  water  in 
square  feet,  and  the  length  in  lineal  feet  of  the  wet  girth  or 
perimeter. 

Then, 
area  in  square  feet  of  the  layer  of  w*ter__rnr  hydraul-c 

Wet  girth  in  lineal  feet 
depth,  so  far  as  this  one  foot  layer  of  water  is  concerned. 

Then  the  velocity  of  flow  of  this  layer  of  water  one  foot 
depth  upon  the  sides  and  bottom  will  be 


In  channels  where  the  actual  depth  of  flow  exceeds  one 
foot,  no  matter  how  greatly,  the  value  of  r  determined  as  above 
will  be  less  than  unity,  but  will  approach  unity.  In  order  to 
err  on  the  safe  side  and  as  a  matter  of  convenience,  we  as- 


226  SULLIVAN'S  NEW  HYDRAULICS. 

eurne  that  r  is  a  constant  equal  unity  in  channels  where  the 
depth  of  flow  is  one  foot  or  greater;  and  under  these  con- 
ditions the  eroding  velocity  or  velocity  of  this  layer  of  water 
is 


<109> 

If  the  channel  is  so  small  that  the  actual  value  of  r  for 
the  whole  volume  of  flow  is  less  than  rr=1.00,  then  the  mean 
velocity  and  all  other  velocities  may  be  considered  as  equal 
and  may  be  found  by  the  formula  for  mean  velocity  in  chan- 
nels of  the  given  degree  of  roughness.  In  either  case  the  ac- 
tual eroding  velocity  will  not  exceed  the  computed  eroding 
velocity,  and  the  computed  result  will  be  a  safe  guide  in  de- 
termining the  grade  of  the  canal. 

89.—  Slope  or  Grade  of  Canal  to  Generate  a  Qiven 
Bottom  or  Eroding  Velocity.—  The  stability  of  the  material 
which  forms  the  perimeter  of  the  canal  must  be  the  controll- 
ing factor  in  determining  the  grade  or  elope  of  the  canal. 
Very  light  soil  will  not  stand  a  bottom  velocity  greater  than 
one  half  foot  per  second  without  serious  erosion,  while  other 
classes  of  soil  will  stand  much  higher  bottom  velocities  with- 
out damage.  When  it  has  been  determined  what  bottom  ve- 
locity is  best  adapted  to  the  material  forming  the  perimeter, 
then  the  slope  or  grade  of  the  canal  (without  reference  to  its 
size)  which  will  be  required  to  generate  that  given  bottom  ve- 
locity will  be 

S=m  v*  ........................................  (110) 

In  which, 

v2=the  square  of  the  proposed  bottom  velocity  in  feet 
per  second. 

m=coefficient  of  velocity  applicable  to  roughness  of  peri- 
meter. 

S=Slope  required  to  generate  the  given  bottom  velocity. 

If  the  channel  is  so  small  that  the  value  of  r  for  the  en- 
tire volume  of  flow  is  less  than  r=i.OO,  then 

S  =  :/7F>  and  tlie  mean  and  bottom  velocities  will  be 
practically  the  same. 


SULLIVAN'S  NEW  HYDRAULICS.  227 

If  the  bottom  velocity  =.J-§L  has  been  decid6dt  then 
the  mean  velocity  for  any  value  of  r  will  equal  the  bottom  ve- 
locity multiplied  by  fr8,  or  v=f/r3X  -/— 

V  m 

The  value  of  m  may  be  selected  from  the  groups  of  data 
of  flow  in  open  channels  heretofore  given. 

90.— Stability  of  Channel  Bed  Materials,— According 
to  the  observations  of  Du  Buat  a  bottom  velocity  of  3  inches 
per  second  will  just  begin  to  work  upon  fine  clay  fit  for  pot- 
tery; a  bottom  velocity  of  6  inches  per  second  will  lift  fine 
sand;  8  inches  per  second  will  lift  sand  coarse  as  linseed;  12 
inches  per  second  will  sweep  along  fine  gravel.  24  inches  per 
second  will  roll  along  rounded  pebbles  an  inch  in  diameter; 
a  bottom  velocity  of  3  feet  per  second  will  sweep  along  shiv- 
ery, angular  stones  as  large  as  eggs.  Professor  Rankine  givea 
the  following  table  of  the  greatest  velocities  close  to  the 
bed  which  are  consistant  with  the  stability  of  the  materials 
mentioned :- 

Soft  clay 0.25  feet  per  second . 

Fine  sand 0.50    "      " 

Course  sand,  and  gravel  as  large  as  peas.  .0.70    •'      "        " 

Gravel  as  large  as  French  beans 1.00    "      "        " 

Gravel  one  inch  diameter 2  25    "      "        " 

Pebbles  1}£  inches  diameter ,3.33    "      " 

Heavy    shingle 4.00    "       "        " 

Soft  rock,  brick,  earthenware ,4»50    "       "        " 

Rock,  various  kinds 6.00    and  upwards. 

See  also  "Civil  Engineer's  Pocket  Book"  by  Trautwine, 
pp.  563,  570,  and  "Irrigation  Engineering"  by  H.  M.Wilson 
page  86,  and  Fanning,  page  622. 

The  experiments  of  Du  Buat  were  in  a  small  wooden 
trough  with  a  smooth  bottom  so  there  was  little  friction  be- 
tween the  moving  particles  of  the  material  and  the  bottom  of 
the  trough.  Loose  material  on  a  smooth  uniform  floor 
would  be  moved  by  a  smaller  bottom  velocity  than  if  it  were 


228  .    SULLIVAN'S  NEW  HYDRAULICS. 

incorporated  in  the  bed  of  an  earthen  channel.  It  is  probable 
that  in  ordinary  earth  the  bottom  velocity  should  be  about 
.70  foot  per  second,  and  the  slope  should  be  S=tn  v*  =  .00031X 
(.70)*  =.0001519. 

91— Adjustment  of  Slope  Or  Grade,  Bottom  Velocities 
and  Side  Slopes  of  Canals,  to  the  Material  Forming  the 
Bed. — In  order  to  preserve  the  efficiency  and  delivery  of  a  ca 
nal,  its  cross-section  must  be  uniform,  sy metrical  and  free  of 
deposits  and  plant  growth.  Caving  and  sliding  banks,  due  to 
the  action  of  frost  upon  side  slopes  steeper  than  the  natural 
angle  of  repose  of  the  material  forming  the  sides  of  the  canal, 
when  such  material  is  reduced  to  powder  by  frost  in  winter 
when  the  canal  is  empty,  not  only  causes  the  filling  up  of  the 
canal,  but  also  leaves  the  banks  rough,  irregular  and  ragged, 
and  greatly  reduces  its  area,  while  it  increases  and  roughens 
the  perimeter.  The  efficiency  or  delivery  of  a  canal  may  be 
reduced  fully  one  third  during  one  winter  from  this  one  cause 
alone.  The  extent  of  damage  thus  done  will  not  be  fully  dis- 
covered until  the  water  has  again  been  admitted  to  the  canal. 
All  the  loose,  disintegrated  material  will  then  be  washed  off 
the  sides  and  deposited  in  the  bottom  in  irregular  heaps. 
These  heaps  will  be  acted  upon  by  the  mean  velocity  in  the 
same  manner  that  a  bridge  pile  or  pier  is  attacked  by  the 
flow,  and  will  thus  be  cut  away  and  redeposited  on  one  side 
where  the  velocity  is  not  sufficiently  great  to  keep  the  mater- 
ial in  suspension  and  in  transit.  This  will  change  the  direc- 
tion of  the  current  to  the  deepest  part  of  the  cross  section 
next  the  opposite  bank  which  produces  an  undercutting  and 
caving  at  that  point  and  a  further  deposit  on  the  side  oppo- 
site the  cutting.  The  thread  of  the  current  is  caused  to  cross 
from  one  side  to  the  other  and  thus  the  energy  of  the  stream 
is  expended  in  destroying  the  banks  and  in  transporting  ma- 
terial from  one  point  to  another.  There  are  few  instances 
in  which  the  bottom  of  a  canal  has  been  scoured  and  eroded 
to  a  serious  extent.  The  silt  and  deposits  nearly  always  come 
from  the  banks  which  clearly  indicates  that  the  side  slopes 


SULLIVAN'S  NEW  HYDRAULICS.  220 

are  too  steep  for  the  material  and  for  the  climate,  or  that  the 
alignment  is  bad,  for  if  the  alignment  IB  bad  and  the  ve- 
locity too  high,  all  the  velocities  are  eroding  velocities  at 
the  bends,  and  consequently  a  very  low  mean  velocity  must 
be  adopted  or  the  banks  must  be  protected  by  paving  or 
otherwise.else  the  annual  expense  of  cleaning  and  repairs  will 
be  excessive.  The  proper  side  slopes  of  a  canal  will  depend 
upon  the  nature  of  the  material  forming  the  perimeter.  The 
side  slope  should  never  be  steeper,  in  climates  subject  to 
frost,  than  the  natural  angle  of  repose  of  the  material  when 
thrown  up  in  considerable  heaps,  loose  and  dry.  In  climates 
subject  to  frost  the  side  slopes  will  be  thoroughly  pulverized 
by  alternate  freezing  and  thawing  when  the  canal  is  empty 
in  winter,  or  above  the  water  level  if  the  water  is  not  turned 
out  in  winter.  Under  these  conditions,  if  the  side  slope  is 
steeper  than  the  natural  angle  of  repose  of  the  material 
when  it  is  perfectly  loose  and  dry,  the  result  is  that  the  ma- 
terial thus  pulverized  by  frost  will  roll  down  into  the  canal 
at  each  thawing  until  the  slope  finally  reaches  its  natural  an- 
gle of  repose  in  a  rough  and  irregular  way.  The  method  of 
determining  the  angle  of  repose  is  not  by  reference  to  pub- 
lished tables  of  such  angles  for  different  materials,  but  by 
throwing  up  a  large  heap  of  the  material  to  be  dealt  with  and 
allowing  it  to  assume  any  angle  it  will.  The  angle  thus  as- 
sumed by  the  sides  of  the  heap  is  as  steep  as  the  side  slopes 
of  the  canal  should  be  in  that  class  of  material.  The  angle 
of  repose  will  be  found  to  vary  widely  for  different  classes  of 
earthy  material,  and  for  most  kinds  the  angle  will  be  much 
steeper  if  the  material  is  damp  or  moderately  wet  than  if  it  is 
either  dry  or  saturated.  Hence  the  angle  should  be  found 
when  the  material  is  perfectly  dry  and  loose. 

The  side  slopes  having  been  made  to  conform  to  the 
angle  of  repose  thus  found,  and  due  attention  having  been 
given  to  the  alignment,  it  is  then  necessary  to  so  adjust  the 
slope  of  the  bottom  of  the  canal  as  to  cause  a  bottom  velocity 
of  flow  most  suitable  to  the  material  of  the  perimeter.  If  the 
canal  is  to  be  of  considerable  width  and  to  have  a  depth  of 


230  SULLIVAN'S  NEW  HYDRAULICS. 

flow  exceeding  one  foot,  then  the  grade  or  slope  should  be 

S=mv8. 

Here  m  is  to  be  selected  from  the  values  of  m  developed 
for  canals  in  like  condition  and  in  like  material,  given  in  the 
groups  of  data  of  flow  in  open  channels. 

The  value  of  v  will  depend  upon  the  bottom  velocity 
which  the  given  material  of  the  perimeter  will  stand  without 
erosion.  The  suggestions  heretofore  (§87)  given  may  assist  in 
determining  what  value  should  be  assigned  to  v  in  the  above 
formula. 

If  the  canal  is  to  be  comparatively  deep  and   narrow,  as 
it  should  be  where  practicable,  then  the  grade  should  be 
mv8        mva 

But  in  this  formula  the  value  of  r  is  found  not  by  taking 
the  quotient  of  the  total  cross-sectional  area  of  the  column  of 
water  by  the  wetted  perimeter,  but  by  assuming  that  there 
is  one  foot  depth  of  water  adhering  to  the  sides  and  bottom, 
the  area  of  which  is  to  be  divided  by  the  total  wet  girth  in 
lineal  feet.  The  resulting  value  of  r  is  that  which  is  to  be 
used  in  determining  the  slope  to  generate  the  given  bottom 
velocity. 

If  the  value  of  r  is  the  true  value  for  total  area  divided 
by  wet  perimeter,  and  v  represents  the  desired  mean  velocity, 
then  the  last  formula  will  give  the  required  slope  to  gener- 
ate the  given  mean  velocity,  without  reference  to  bottom 
velocity. 

In  very  light  soil  mixed  with  fine  sand  the  action  of  waves 
will  reduce  the  side  slopes  much  flatter  than  the  angle  of  re- 
pose of  the  material  when  dry  or  only  damp.  If  fluming,  pud- 
dling, or  paving  cannot  be  resorted  to  where  the  canal  passes 
through  such  material,  then  the  canal  should  have  a  cross- 
section  elliptical  in  form,  and  the  bottom  or  scouring  velocity 
should  not  exceed  .45  foot  per  second,  and  great  care  must  be 
taken  to  avoid  bad  alignment. 

The  grade  of  the  canal  having  been  determined  with 
reference  to  the  greatest  bottom  velocity  the  material  of  the 


SULLIVAN'S  NEW  HYDRAULICS  231 

bed  will  safely  stand,  it  then  becomes  necessary  to  determine 
the  dimensions  of  the  canal  with  that  given  grade  which  will 
cause  the  discharge  or  carriage  of  the  required  quantity  of 
water. 

92.— Dimensions  of  Canals  to  Carry  Given    Quantities. 

—In  the  case  of  canals  with  side  slopes  of  about  2  horizontal 
to  1  vertical,  and  of  considerable  capacity,  the  value  of  the 

hyd  aulic  mean  depth  —,  may  be  approximately  found  by 
formula  (64)  which  is 


In  this  connection  see  §§  19  and  3.  The  required  value  of  r 
being  thus  found  in  terms  of  cubic  feet  per  second  q,  then, 

a=r8Xl2-566*»  and  wet  perimeter,  P=_a_.  For  reasons  here- 
tofore pointed  out  these  formulas  are  not  generally  applicable 
to  all  forms  of  cross-section  and  capacities  of  open  channels, 
and  when  the  values  of  a,  p,  and  r  have  been  calculated  in 
this  manner,  the  general  formula  for  velocity  should  be  ap- 
plied as  a  check.  When  the  mean  velocity  is  thus  found, 
then  q=aXv. 

For  example  suppose  the  grade  decided  upon  for  a  canal 
is  S=.0002754=l  in  3631.08,  and  the  value  of  m  applicable  to 
the  class  of  gravelly  earth  is  m=.00034.  What  area  in  square 
feet  and  what  wet  perimeter  and  what  value  of  r  would  be 
required  to  cause  the  canal  to  discharge  1,000  cubic  feet  per 
second,  the  side  slopes  being  2  to  1?  In  the  first  place  find  the 
required  value  of  r  by  formula  (64)  which  will  be  r=5.121. 

Then  required  area  in  square  feet,  a=raX12.5664=329.554. 

The  required  wet  perimeter  =JL=  32^f    =64.353. 
r  5.121 

Taking  33.1668  feet  of  the  wet  perimeter  as  the  bottom 
width  of  the  canal,  there  will  have  to  be  a  depth  at  center 
sufficient  to  take  up  the  remaining  31.1862  feet  of  wet  peri- 
meter which  is  to  be  divided  equally  between  the  two  side 


232  SULLIVAN'S  NEW  HYDRAULICS. 

elopes.      Then  the  wet   perimeter  of  one  side  slope  will  be= 
31.1862 


As  the  side  slopes  are  2  horizontal  to  1  vertical,  a  verti- 
cal depth  of  water  equal  about  one  half  the  length  of  one 
side  slope,  or  about  7  feet  in  this  case,  will  be  required.  So 
making  the  depth  of  water  at  the  center  equal  7  feet,  and  the 
bottom  width  as  above,  equal  33.1668  feet,  and  the  side  slopes 
2  to  1,  we  have  the  length  of  one  side  slope  =-/  7*  +14*  =15.65 
feet.  Then  total  wet  perimeter  =15.65+15.65-(-33.1668= 
64.466  feet. 

The  actual  area  will  be  330.1676  feet.    The  actual  value  of 

r  will  be  =  330-1676  =5.121.     Now  as  a  check  on   this  calcu- 
04.460 

lation  we  must  apply  the  general  formula   for  mean  velocity 
to  the  slope    and    dimensions   above    found,  and    we    have 

.064.    And  the 

quantity  in  cubic  feet  per   second  which  will  be  discharged 
will  be  q=areaXvelocity=330.1  676X3.064=1011.63  cubic  feet, 

,-Q 

Tho  bottom  velocity  in   this  canal   would  be  v=.J—  = 

Vm 


While  it  is  seen  that  the  dimensions  of  a  canal  of  this 
form  of  cross  section  and  capacity  ruaybe  closely  ascertained 
by  the  formulas  for  r,  a  and  p,  as  above  shown,  yet  these 
particular  formulas  do  not  apply  to  small  canals  nor  to  rec- 
tangular canals,  with  any  degree  of  accuracy.  These  parti- 
cular formulas  do  apply,  however,  with  exactness  to  pipes  or 
circular  closed  channels  running  full. 

93.— Allowance  In  Cross  Section  of  Canals  For  Leak- 
age and  Evaporation. — The  amount  of  loss  by  leakage  and 
evaporation  from  a  canal  will  depend  upon  the  climate,  the 
nature  of  the  soil,  the  length  of  the  canal,  the  depth  of  flow, 
and  above  all  the  position  of  the  canal  with  reference  to  the 


SULLIVAN'S  NEW  HYDRAULICS  233 

elevations  and  depressions  of  the  surface  of  the  surrounding 
country. 

If  the  canal  is  constructed  upon  the  highest  line  of  the 
land  through  which  it  passes,  the  leakage  from  it  will  be 
great,  and  because  of  its  elevated  position  it  can  never  regain 
any  part  of  this  loss  by  return  seepage.  Such  location  also 
exposes  the  water  surface  to  the  action  of  the  sun  and  wind, 
and  thus  large  losses  occur  by  evaporation,  especially  if  the 
canal  is  wide  and  shallow.  In  arid  regions  where  irrigation 
is  not  general  and  abundant,  the  sub-surface  water  level  i<3  at 
considerable  depth  below  the  surface,  but  after  irrigation  has 
been  practiced  for  several  years,  the  earth  becomes  saturated 
and  the  sub-surface  water  level  rises  near  to  the  surface.  Un- 
til this  occurs  the  loss  from  new  canals  in  such  regions  will 
be  very  great.  After  irrigation  has  been  practiced  for  a  num- 
ber of  years,  and  has  become  general  in  the  given  locality,  the 
canals  situated  along  side  hills  and  skirting  the  valleys  will 
gain  vastly  more  by  seepage  into  the  canal  than  will  be  lost 
by  leakage  and  evapoiation  combined.  In  some  canals  in 
Colorado  the  gain  by  seepage  into  the  canal  is  as  great  as 
two  thirds  the  total  original  quantity  admitted  into  the  canal 
at  its  head.  This  occurs  only  in  canals  located  where  irri- 
gation has  been  practiced  for  years,  and  in  canals  so  situated 
on  side  hills  or  along  the  foot  of  the  hill,  as  to  admit  of  the 
seepage  flowing  into  the  canal. 

The  loss  by  leakage  and  evaporation  from  new  canals  in 
arid  regions  varies  from  20  to  75  per  cent  of  the  quantity  ad- 
mitted into  the  canal,  according  to  the  nature  of  the  soil  and 
the  length  of  the  canal.  As  the  canal  becomes  silted  and  the 
sub-surface  water  level  rises,  the  leakage  will  decrease,  and  if 
the  canal  is  so  located  as  to  admit  of  it,  the  gain  by  return 
seepage  will,  in  the  course  of  a  lew  years,  more  than  balance 
the  loss  by  leakage  and  evaporation. 

In  regions  where  the  rainfall  is  great  it  is  probable  that 
the  seepage  into  a  new  canal  will  offset  the  leakage  from  the 
first  opening  of  the  canal,  because  the  sub-surface  water  level 
is  already  very  close  to  the  surface  of  the  ground. 


234  SULLIVAN'S  NEW  HYDRAULICS. 

In  making  allowance  in  cross-sectional  area  of  a  canal 
to  cover  these  losses,  it  should  be  by  way  of  extra  depth. 

94.— Where  a  Flume    Forms  Part  of  a  Canal.— Where 

the  course  of  a  canal  would  pass  around  on  a  very  steep  side 
hill,  or  through  stretches  of  very  porous  earth,  or  across  low 
depressions,  flumes  are  frequently  adopted  as  portions  of  the 
canal  for  such  reaches.  In  this  event  the  question  arises  as 
to  the  proper  ratio  of  flume  cross  section  to  that  of  the  canal, 
of  which  the  flume  forms  a  part.  The  determination  of  this 
question  involves  a  consideration  of  the  relative  degree  of 
roughness  of  the  two  classes  of  channel,  and  the  difference  in 
slope  or  grade  of  the  flume  and  the  canal,  as  well  as  the  length 
of  the  flume  and  its  alignment.  If  the  flume  is  short  and 
upon  the  same  grade  as  that  of  the  canal,  and  has  no  vertical 
fall  at  its  lower  end,  the  water  will  not  acquire  a  velocity  in 
such  short  flumes  much  greater  than  that  in  the  canal,  and 
therefore  the  area  of  the  flume  under  such  conditions  cannot 
be  reduced  much  below  that  of  the  wetted  area  of  the  canal. 
While  the  velocity  of  flow  will  usually  be  greater  in  a  flume 
than  in  a  canal  of  equal  slope,  yet  at  the  entry  to  the  flume 
the  water  has  only  the  velocity  of  the  canal,  and  the  head  due 
to  that  velocity.  It  must  flow  a  sufficient  distance  in  the 
flume  to  acquire  the  greater  velocity  due  to  the  smoother  peri- 
meter before  the  depth  and  area  of  the  flume  can  be  materi- 
ally reduced  from  that  of  the  connecting  canal,  otherwise 
there  will  be  an  overflow  at  the  upper  junction  of  the  flume 
with  the  canal.  The  flume  should  converge  from  the  mean 
width  oi  the  canal  at  the  junction,  to  the  standard  section 
adopted  for  the  flume,  in  a  length  varying  from  50  to  200  feet 
according  to  the  difference  in  slope  and  in  roughness  of  the 
flume  and  the  canal.  The  value  of  C  might  be  56  for  the  canal 
and  anywhere  from  70  to  130  for  the  flume,  according  to 
the  method  and  materials  adopted  in  its  construction  and 
alignment. 

A  straight  canal  in  firm,  dense  earth  and  in  best  condi- 
tion develops  C—  75.00,  while  a  rough,  crooked  flume  with 
battens  on  the  inside  develops  C— 70.00.  In  such  cases  as 


SULLIVAN'S  NEW  HYDRAULICS,  235 

this  the  flume  would  require  an  area  slightly  in  excess  of 
that  of  the  canal,  or  would  require  an  equal  area  and  steeper 
grade.  On  the  other  hand  the  value  of  C  for  a  rough  canal 
may  be  as  low  as  40,  while  the  value  of  C  for  a  very  smooth 
well  jointed  hard  wood  flume  of  good  alignment  might  be  as 
high  as  130. 

The  slopes  being  equal,  the  velocities  will  be  as  f/r*  in 
the  one  is  to  J/r3  in  the  other,  as  modified  by  the  respective 
values  of  C,  or  viviiCJ/r^CJ/r3.  If  the  slopes  are  different 
then  v:v::  C£/rVS:Cf/rVS 

The  value  of  C  may  be  taken  from  the  data  of  like 
flumes  and  channels  given  in  the  groups,  Chapter  2. 

95— Mean  Velocity  In  Uniform  Sections  of  Canals 
Found  by  Floats. 

In  straight  sections  of  canals  of  uniform  cross-section 
where  the  thread  of  the  greatest  velocity  is  midway  between 
banks  and  just  beneath  the  water  surface,  the  place  of  mean 
velocity  will  be  found  at  .50  of  total  depth  at  a  point  midway 
between  the  center  of  the  canal  and  the  bank,  unless  the 
depth  of  flow  is  less  than  two  feet,  in  which  case  the  place 
of  mean  velocity  will  be  at  or  just  above  mid-depth  at  a  point 
midway  between  the  bank  and  the  middle  of  the  canal,  as- 
Burning  that  the  sides  and  bottom  of  the  canal  are  fairly 
smooth.  In  shallow  canals  with  gravel  and  pebbles  along 
the  bottom  the  place  of  mean  velocity  is  very  near  mid-depth, 
aometimes  slightly  above,  and  at  one-fourth  the  width  of  the 
canal  from  the  bank.  A  large  tin  bucket  loaded  with  gravel 
and  covered,  may  be  suspended  by  a  fine  wire  at  this  depth 
and  connected  to  a  flat  circular  float  on  the  surface  no  larger 
than  is  absolutely  necessary  to  support  the  submerged  bucket 
at  proper  depth.  This  double  float  is  to  be  placed  at  some 
distance  above  the  upper  end  of  a  measured  length  of  the 
canal,  and  adjusted  to  proper  position  with  reference  to  the 
bank  or  width  of  the  canal,  and  with  reference  to  depth,  and 
allowed  to  travel  over  the  given  course  a  number  of 
times.  The  average  time  required  for  its  passage  over  the 
given  number  of  feet  length  of  the  canal  will  closely  approxi- 


236  SULLIVAN'S  NEW  HYDRAULICS. 

mate  the  rate  of  mean  velocity.  The  difficulty  of  ascertain- 
ing the  exact  number  of  seconds  which  elapse  between  the 
time  the  float  crosses  the  line  at  the  upper  station  and  arrives 
exactly  at  the  line  of  the  lower  station,  will  probably  cause  a 
slight  error  in  the  final  determination  of  the  mean  velocity. 
For  this  reason  the  measured  course  should  be  several  hun- 
dred feet  in  length.  If  the  channel  is  rough  and  winding  the 
float  will  be  cast  either  too  near  the  bank  or  into  mid-cur- 
rent, and  the  result  is  without  value.  Float  measurement  of 
mean  velocity  is  practicable  only  in  channels  of  uniform 
width  and  depth.  The  surface  velocity  has  no  particular  re- 
lation to  the  mean  velocity,  and  it  is  therefore  impossible  to 
deduce  the  mean  from  the  surface  velocity.  The  ratio  be- 
tween surface  and  mean  velocity  varies  with  the  form  of 
cross-section,  roughness  of  perimeter,  uniformity  of  cross- 
section,  variation  in  slope,  depth  of  flow  and  hydraulic  radius 
and  alignment  of  the  channel. 

The  surface  velocity  depends  mainly  on  the  slope,  while 
the  mean  velocity  depends  upon  the  value  of  {/r3as  well  as  up- 
on the  roughness  and  slope  of  the  channel.  In  rough,  stony 
channels  of  varying  cross-section  and  small  depth  of  flow 
there  is  scarcely  any  difference  between  surface  and  mean 
velocity. 


CORRECTION  OF  TEXT. 


It  is  probable  that  no  one  ever  turned  his  manuscript 
over  to  the  printer  without  a  lively  sense  of  its  probable  de- 
merits when  it  shall  stare  one  in  the  face  from  the  printed 
page. 

The  greater  part  of  the  book  was  written  several  years 
ago.  and  portions  of  it  were  published  in  various  journals  in 
1894  and  1895,  While  the  ultimate  conclusions  reached  and 
formulas  deduced,  as  appear  in  the  text,  are  correct,  yet  some 
of  the  reasoning  is  at  fault,  and  not  clear.  The  author  would 
be  glad  to  stop  the  press  and  re -write  the  entire  book  after 
having  seen  half  the  printed  "proof,"  but  it  ie  too  late. 

He  must  therefore  resort  to  the  alternative  of  writing 
a  criticism  of  his  own  work,  and  thus  forstall  the  other 
fellow. 


The  three  important  principles  which    are  sought   to   be 
established  are:- 

(I)— That  it  is  the  effective  value  of  the  head  or  slope  which 
varies  with  some  function  of  the  diameter  or  hydraulic 
mean  radius,  or  mean  depth,  and  not  the  coefficient 
that  varies. 

(II)— That  for  any  given  class  of  wet  perimeter,  or  any  given 
degree  of  roughness,  the  coefficient  is  necessarily  a  con- 
stant for  all  heads,  slopes,  velocities,  diameters  or  mean 
hydraulic  radii. 

(Ill)— That  the  value  of  the  coefficient  is  governed  absolutely 
by  the  roughness  of  wet  perimeter,  and  by  nothing  else, 
and  is  therefore  an  absolutely  reliable  index  of  the 
roughness  of  perimeter. 

FIRST  PROPOSITION. 

That  the  Effective  Value  of  a  Constant  Head  or  Slope 
Varies  With  Some  Function  d/d8,  or   /R8)  of  the  Diam- 


238  SULLIVAN'S  NEW  HYDRAULICS. 

eter,  or  of  the  Hydraulic  Mean  Radius,  and  that  the  Coe* 
ficient  does  not  Vary  with  the  Diameter  or  Hydraulic  Mean 
Radius  at  all. 

If  a  series  of  pipes  or  open  channels  of  exactly  equal 
roughness  of  perimeter,  but  of  different  diameters,  or  differ- 
ent hydraulic  mean  radii,  have  exactly  the  same  head  or 
slope  per  foot  length,  it  ia  well  known  that  the  pipe  having 
the  greatest  diameter,  or  the  open  channel  having  the  great- 
est hydraulic  mean  depth  (R),  will  generate  the  greatest 
velocity  of  flow,  and  the  pipe  having  the  least  diameter,  or 
the  open  channel  having  the  least  mean  hydraulic  depth,  will 
generate  the  least  velocity  of  flow.  As  all  these  pipes,  or  all 
these  channels,  are  of  equal  roughness,  and  all  have  exactly 
equal  heads  or  slopes,  it  is  evident  that  the  velocity  would  be 
the  same  in  each  of  them  if  the  constant  head  or  slope  were 
not  made  more  effective  with  an  increase  in  diameter  or  hy- 
draulic mean  depth.  This  being  true,  the  next  inquiry  is, 
what  is  the  ratio  of  increase  in  the  effectiveness  of  the  given 
head  or  slope  as  diameter  or  hydraulic  mean  depth  increases? 

To  solve  this  problem  we  must  appeal  both  to  the  laws 
of  friction  or  resistance,  and  of  gravity.  The  resistance,  or 
head  lost  by  resistance,  will  be  directly  as  the  roughness  of 
perimeter,  and  directly  as  the  extent  of  perimeter,  and  also 
directly  as  the  square  of  the  velocity. 

As  demonstrated  in  the  text  the  wet  perimeter  or  extent 
ot  friction  surface,  varies  exactly  with  d  or  r.  (See  pp.  3£ 
36,39,40.) 

But  if  there  were  no  friction  or  resistance,  then  the  velocity 
would  be  the  same  for  the  same  actual  slope  regardless  of  the 
value  of  d  or  r.  While  the  friction  surface  and  consequently 
the  absolute  loss  of  head  by  resistance,  increases  only  as  d  or 
r,  the  cross  section  of  the  column  of  water  increases  as  d8  or 
r»,  or  as  the  sectional  area. 

The  absolute  head  or  slope  therefore  increases  as  the  area, 
or  as  d8  or  r2,  while  the  absolute  loss  of  head  increases  only 
as  d  or  r.  It  is  evident  then,  that  the  absolute  head  or  slope, 
which  varies  as  d8  or  r*,  must  be  modified  by  the  absolute 
loss  of  head  or  slope  which  varies  as  d  orr.  Then  the  mean. 


SULLIVAN'S  NEW  HYDRAULICS.  239 

head,  or  relative  head,  of  all  the^  particles  of  water  in  the 
croBB  section  will  vary  with  d*  as  modified  byd,  or  with  r1  as 
modified  by  r.  As  r*  must  not  be  increased  by  r,  but  must  be 
modified  by  i,  we  must  reduce  both  d  and  d»,  or  r  and  r1,  in 
the  same  ratio,  in  order  to  obtain  a  reducing  or  modifying 
multiplier.  To  accomplish  this  result,  we  say  that  j/d  bears 
the  same  relation  to  d  that  d  bears  to  d2,  or  that  y/r  bears 
the  same  relation  to  r  that  r  bears  to  r*.  In  other  words  to 
maintain  the  ratio,  of  r  to  r*,  or  d  to  d»,  and  at  the  same  time 
obtain  a  multiplier  which  will  give  the  combined  net  effects 
of  d  and  d2.  or  r  and  rs,  upon  the  value  of  H  or  S,  it  is  neces- 
sary to  take  the  square  root  of  both  d  and  d8,  or  of  both  r 
and  r*.  We  then  say  that,  relatively,  the  area  or  absolute 
head  (d"  or  r2)  varies  with  y'd*=J,or  with  ^/r*=r,  while  the 
friction  surface  or  absolute  loss  of  head  varies  with  -/d  or  y'r. 
and  consequently  the  relative  mean  head  of  all  the  particles  in 
the  cross  section  will  vary  with  the  resultant  of  these  two 
effects,  which  will  be  as  d^/d,  or  as  R^/R.  Thus  we  obtain 
the  modifying  multiplier  j/d,  or  y/r,  while  we  maintain  the 
correct  ratio  of  friction  surface  to  area,  or  of  loss  of  head 
to  gain  in  head  as  d  or  r  varies  for  a  constant  head  or 
slope. 

It  is  evident  then  that  the  constant  head  or  slope  be- 
comes more  effective  or  less  effective  as  dy/d=>/d8,  or  \/r*, 
increases  or  decreases. 

SECOND  PROPOSITION. 

That  for  any  Given  Degree  of  Roughness  of  Wet  Peri- 
meter, the  Coefficient  is  a  Constant  for  all  Heads,  Slopes, 
Velocities,  Diameters  or  Hydraulic  Mean  Depths. 

It  was  shown  in  the  foregoing  discussion  that  the  effect- 
ive value  of  the  head  or  slope  varies  with  y/d3ory/r3.  By  the 
law  of  gravity  the  square  of  the  velocity  must  always  be  pro 
portional  to  the  head  or  slope  in  any  given  pipe  or  channel, 
or  va=2gH.  As  a  necessary  consequence  of  this  law,  it  is 
obvious  that  anything  which  affects  the  effective  value  of  the 
head  or  slope  must  at  the  same  time  equally  affect  the  value 
of  vs. 


240  SULLIVAN'S  NEW  HYDRAULICS. 

When  we  write  m=— ^3 —     and     remember      that     the 

effective  value  of  S  increases  with  i/r*,  and  that  any  increase 
in  the  effective  value  of  S  must  a'so  increase  v»  in  the  same 
ratio,  it  is  evident  that  as  both  dividend  and  divisor  increase 
aliKb  the  quoti?nt,  m,  will  continue  a  constant  for  all  values 
of  r,  Sand  v*.  Their  relation  is  such  that  we  cannot  increase 
the  effective  value  of  S  without  also  increasing  the  value  of 
v1  in  the  same  ratio.  Hence  m  is  necessarily  a  constant. 

THIRD  PROPOSITION. 

That  the  Value  ot  the  Coefficient  is  Governed  Abso- 
lutely by  the  Roughness  ot  the  Wet  Perimeter,  and  by 
Nothing  Else,  and  is  Consequently  an  Absolutely  Corrrect 
Index  of  the  Roughness. 

When  we  inspect  the  formula  for  the  coefficient,  m=    y8  • 

it  is  apparent  that  m  is  simply  the  expression  for  the  ratio  of 
effective  slope  to  the  square  of  the  velocity.  If  the  pipe  or 
channel  is  rough  it  will  require  a  large  value  of  the  effective 
slope,  Sy/r8,  to  generate  a  small  value  of  v*.  Consequently 
the  ratio,  m,  of  effective  slope  to  v*,will  be  large  in  rough 
channels.  But  if  the  channel  is  uniform  in  area,  and  smooth 
then  a  small  effective  value  of  slope,  Sy'r3,  will  generate  a 
relatively  large  value  of  vs,  and  hence  the  ratio,  m,  will  be 
small  for  smooth  perimeters.  As  m  is  simply  the  expression 
for  this  ratio,  and  as  this  ratio  depends  exclusively  on  the 
roughness  of  perimeter,  it  is  obvious  that  m  will  vary  only 
with  the  roughness. 

The  coefficient,  C=AI    y2    ,  is  simply  the  square  root   of 

VSyr8 

the  reciprocal  of  m,  and  will  consequently  be  a  constant,  like 
m,  for  any  given  degree  of  roughness.  But  being  the  square 
root  of  the  reciprocal  of  m,  C  will  vary  with  the  roughness  in 
the  exact  opposite  way  from  m— that  is,  C  will  be  large  for 
smooth  perimeters  and  small  for  rough  perimeters,  while  m 
will  be  large  for  rough  perimeters  and  small  for  smooth  peri- 
meters. 


SULLIVAN'S  NEW  HYDRAULICS.  241 

As  either  of  the  coefficients  vary  only  with  the  rough- 
ness of  wet  perimeter,  but  is  very  sensitive  to  uny  change  in 
roughness,  it  will  be  found  that  C  will  decrease  as  depth  of 
flow  increases  in  all  channels  where  the  sides  are  rougher 
than  the  bottom,  and  will  increase  with  increase  of  depth  of 
flow  in  all  channels  where  the  sides  are  smoother  and  more 
uniform  than  the  bottom.  In  other  words  C  will  vary  as  the 
mean  of  the  roughness  varies.  See  in  this  connection  §13 
page  58,  and  also  p  p.  27,  28,  29,  41,  42. 

The  best  form  of  the  formula  for  general  use  is, 


This  form  of  the  formula  also  shows  by  mere  inspection 
that  the  effective  value  of  S  varies  with  ^/r*. 

If  the  formula  is  written,  v=C  Vr*  l/S,  the  actual  result 
would  be  the  same  whether  we  say  that  C  or  \/S  varies  with 
*  V  r3,  but  as  C  insists  on  being  constant,  it  is  evident  that  it 
is  the  effective  value  of  S  that  varies  with  v/r8,  and  the  writer 
desires  to  correct  all  statements  to  the  contrary,  It  is  some- 
what absurd  to  insist  that  the  coefficient  is  a  constant  and  at 
the  same  time  to  claim  that  it  varies.  The  coefficient  in  our 
formula  can  vary  only  as  the  average  of  roughness  of  the  en- 
tire wet  perimeter.  In  the  Chezy  or  Kutter  form 
of  formula,  the  coefficient  must  vary  as  the  roughness 
and  also  as  {/r.  (See  pp.  6,  7,  42,  44.)  Hoping  that  this  ab- 
surdity is  fully  corrected  in  this  explanatory  note,  and  asking 
pardon  for  having  committed  such  a  glaring  fault,  the  author 
commits  the  work  to  the  hands  of  the  profession  with  the 
further  hope  that  its  merits  may  outweigh  its  faults. 

MARVIN  E.  SULLIVAN. 
Longmont,  Colorado, 

November,  1st,  1899. 


APPENDIX  1 


Suggestions  Relating  to  Weir  and   Orifice  Measurements  of 
Flowing   Water. 

96— Remarks   in  Relation  to  Weir  Coefficients.— In  the 

third  remark  under  Group  No.  2  §  14,  a  general  form  of  Weir 
formula  was  suggested.  It  is  not  here  intended  to  discuss  the 
well  known  theory  of  flow  over  measuring  weirs  with  sharp 
crests  and  full  or  partial  contraction,  any  further  than  to 
point  out  what  the  writer  believes  would  be  an  improved 
method  of  application  which  is  believed  would  reduce  the 
errors  in  such  determinations.  From  the  nature  of  a  meas- 
uring weir  it  is  impossible  that  the  head  or  depth  upon  the 
weir  should  ever  be  great,  and  consequently  the  velocities  are 
never  very  high,  even  in  the  cases  where  there  is  velocity  of 
approach.  The  amount  of  resistance  to  flow  (being  as  v«) 
offered  by  the  edges  or  perimeter  of  the  notch  is  therefore  a 
small  factor  in  the  sum  total  of  the  coefficient  of  discharge. 
The  important  factor  is  the  coefficient  of  contraction.  It  is 
usual  to  combine  the  coefficient  of  resistance  with  the  coeffi- 
of  contraction  and  their  product  forms  the  coefficient  of  dis- 
charge, which  is  usually  assigned  a  mean  value  of  .62.  For 
the  reason  that  these  two  independent  coefficients  which 
combined  form  the  usual  weir  coefficient  of  discharge,  do  not 
vary  in  the  same  manner  under  similar  conditions,  it  has 
been  found  necessary  to  find  their  combined  value  for  each 
given  depth  upon  the  weir  and  for  each  given  length  of  notch, 
and  for  each  form  of  notch.  If  the  length  of  weir  notch  re- 
mains constant,  a  small  change  in  depth  upon  the  weir  will 
greatly  affect  the  value  of  the  combined  coefficient,  or  coeffi- 
cient of  discharge.  This  cannot  be  attributed,  except  in  very 
small  part,  to  the  resistance  at  the  edges  of  the  notch,  for 
a  small  change  in  depth  upon  the  weir  does  not  greatly  affect 
the  ratio  of  area  to  perimeter  of  the  notch,  which  may  be 


SULLIVAN'S  NEW  HYDRAULICS.  24:'. 

regarded  as  a  very  small  fractional  length  of  open  channel. 
The  effect  upon  the  combined  coefficient  of  varying  the  depth 
upon  the  weir  must  therefore  be  accounted  for  in  the  factor 
representing  contraction  of  the  discharge.  It  is  evident  from 
the  discussion  of  coefficients  of  flow  in  pipes  and  open  chan- 
nels (§  §  3  to  7)  that  the  resistance  to  flow  offered  by  the  edges 
of  the  notch  will  vary  as  H  and  \/ra.  But  the  coefficient  of 
contraction  which  is  the  controlling  and  important  factor  has 
no  known  relation  to  the  value  of  r.  The  coefficient  of  contrac- 
tion is  affected  greatly  by  the  position  of  the  weir,  the  depth 
upon  the  weir,  the  distance  from  the  crest  to  the  bottom  of  the 
channel,  the  distance  between  the  shoulders  of  the  notch  and 
the  banks  of  the  channel,  and  the  velocity  of  flow  through  the 
notch. 

The  experiments  of  Mr.  J.  B.  Francis  upon  the  same  weir 
of  constant  length,  and  where  all  conditions  were  constant  ex- 
cept the  depth  upon  the  weir,  show  that  a  change  of  depth 
alone  upon  any  given  sharp  crested  weir  of  the  usual  form 
will  greatly  affect  the  value  of  the  coefficient  of  discharge, 
and  further  show  that  the  variations  of  the  coefficient  of  con- 
traction apparently  follow  no  law.  The  coefficient  will  de- 
crease as  depth  increases  until  a  certain  depth  is  reached 
(depending  upon  the  proportions  of  the  notch)  and  then  in- 
creases with  a  further  increase  in  depth  up  to  a  certain  point 
where  it  will  again  begin  to  decrease  to  a  small  extent  until 
it  becomes  nearly  constant  for  great  depths  (if  such  were 
practicable). 

To  make  the  usual  weir  coefficients  apply  with  any  de- 
gree of  accuracy  is  not  a  simple  matter  by  any  means,  for  the 
conditions  must  be  identical  with  those  under  which  the 
given  coefficient  was  determined,  The  ratio  of  area  of  notch 
to  area  of  channel,  the  depth  or  height  of  overfall,  the  height 
of  crest  above  the  bottom  of  the  channel  on  the  upstream  side 
of  the  weir,  the  position  of  the  weir,  whether  at  right  angles 
to  the  thread  of  the  channel,  and  vertical,  and  rigidly  straight 
or  allowed  to  bend  under  pressure,  all  affect  the  coefficient  of 
contraction,  in  addition  to  the  influence  of  varying  the  depth 
upon  the  weir.  There  are  so  many  different  influences  bear- 


244  SULLIVAN'S  NEW  HYDRAULICS. 

ing  upon  the  coefficient  of  contraction  that  we  can  never  be 
certain  of  its  value  except  under  given  favorable  conditions 
which  do  not  often  occur  in  actual  practice.  It  is  therefore 
suggested  that  it  would  be  safer  practice  where  careful  de- 
terminations are  to  be  made  to  avoid  all  these  uncertainties 
by  suppressing  all  contraction.  When  this  is  done  there  re- 
mains only  the  coefficient  of  resistance  of  the  edges  of  the 
notch  to  be  dealt  with,  and  the  law  of  its  variation  is  known. 
In  order  to  suppress  contraction  it  is  suggested  that  the 
notch,  whether  rectangular,  triangular  or  trapezoidal,  should 
be  chamfered  on  the  upstream  side  of  the  notch  to  the  form 
of  the  vena  contracta^instead  of  placing  the  chamfered  side 
downstream.  As  illustrating  the  desultory  manner  or  vari- 
ation of  the  coefficient  of  discharge  of  a  sharp  crested  weir 
the  first  three  columns  I,  H,  and  q,  quoted  by  Fanning  from 
Francis' experimental  data  (Table  68,  page  288  Water  Supply 
Engineering)  are  given  in  the  following  table,  and  the  column 

v  was  computed  by  the  formula    v=-^L,  and  from  these  data 

the  resulting  values  of  m  were  computed, 

The  fundamental  formula  for  flow  over  weirs   with  sharp 
crests  may  be  written 

v=%-\l— - — >  or  v—  m  %V  2gH=5.35  m^/H. 
V    m 

Whence 

m=   28-6225H  ^  .f  m  is  U8ed  as  adiviBor> 

v* 
Or 

m= - =-  /      v ,  if  m  is  used  as  a  multiplier. 

5.35^      V  28.6225H  ' 

,  or  q=AreaX5.35  m^E 


SULLIVAN'S  NEW  HYDRAULICS. 
TABLE.  No.  41 — Table  of  Weir  Data. 


245 


L 

Feet 

H 

Feet 

Cubic 
Feet 
Sec. 

A 

Feet 

V 

Feet 
Sec. 

V2 

Feet 
Sec. 

R 
Feet 

Coefficient 
"5.35T/H 

9.997 
9.997 
9.997 
9.997 

0.62 

S:i 

1.56 

16.2148 
23.4304 
45  5654 
626019 

6.198 
7.997 
12.496 
15595 

2.610 
2.929 
3.648 
4014 

6.8121 
8.5790 
13.2875 
161122 

0.5515 
0.6900 
1.00°0 

1.188U 

.6195 
.6121 
.6046 
.6007 

L=  length  in  feet  of  notch. 
H=  depth  in  feet  upon  the  weir. 
q=  cubic  feet  per  second  actually  discharged. 
A=LXH=Area  in  square  feet=depth  of  water  upon  the 
irXlength  of  notch. 

v=-S-=mean  velocity  in  feet  per  second. 
R=  hydraulic  radius  in  feet  of  notch=-2_ 


m=Coefficient  of  discharge: 


v 
5.35/H' 

In  these  experiments  the  conditions  all  remained  con* 
stant  except  the  depth  H,  upon  the  weir. 

In  the  formula 
v=5.35  m  v/H 

if  we  combine  the  value  of  m  with  the  constant  5.35=%\/2g, 
the  following  values  of  the  coefficient  C  result:- 

fl=.62,  m=.6195,  5.35Xm=C=3.3143. 

H=.80,  m=.6121,  5.35Xm=C=3.2747. 

H=1.25,  m=.6096,  5.35Xm=C=3.2613. 

H=1.56,  m=.6007,  5.35Xm=C=3.2137. 

Whence, 

q=AreaX<V  H=C(LX  H)/H 

It  is  evident,  even  for  different  depths  upon  the  same 
weir,  that  if  the  constant  value  C=3.33  is  used,  the  results 
must  be  erroneous.  Suppose  the  velocity  of  approach  is  consid- 
erable, as  on  mountain  streams,  and  that  the  weir  notch  (rec- 
tangular) is  nearly  as  long  as  the  stream  is  wide,  as  often 


246  SULLIVAN'S  NEW  HYDRAULICS. 

becomes  necessary,  then  the  eloping  banks  will  approach  the 
submerged  corners  of  the  notch  and  greatly  affect  the  coeffi- 
cient  of  contraction,  but  to  what  extent,  is  merely  surmise. 
It  is  frequently  the  case  that  in  order  to  stop  the  leaks  under 
and  around  the  weir,  earth,  straw  and  brush  are  banked 
against  its  upper  side,  thus  training  the  flow  upon  the  notch 
and  also  preventing  full  contraction.  This  affects  both  the 
real  value  of  H  or  vz  and  the  contraction  of  the  discharge. 

The  range  of  experimental  coefficients  as  determined  by 
Francis  was  very  small,  being  mostly  for  weirs  about  10  feet 
length  with  depth  upon  the  weir  varying  from  about  six 
inches  to  1.60  feet.  The  variation  of  the  coefficient  of  contrac- 
tion was  found  so  fitful  and  irregular  as  the  ratio  of  length  to 
depth  was  changed  and  with  different  depths  upon  any  given 
length  of  weir,  that  Mr.  Francis  advised  caution  in  the  appli- 
cation of  his  formula  and  coefficients  in  cases  not  falling  di- 
rectly within  the  experimental  conditions.  It  is  assumed  that 
the  contraction  of  the  discharge  over  a  sharp  crested  weir  in 
full  contraction  is  analogous  to  the  contraction  of  the  jet 
from  a  sharp  edged  orifice  in  thin  plate.  If  the  numerous 
tables  of  experimental  orifice  coefficients  determined  under 
various  heads  above  the  center,  and  under  various  proportions 
of  height  to  width  of  orifice  be  investigated,  it  will  be  found 
that  each  form  of  orifice,  or  each  ratio  of  height  to  width, 
develops  a  distinct  series  of  values  of  the  coefficient  as  the 
head  varies. 

The  coefficient  for  an  orifice  will  either  decrease  or  in- 
crease with  the  head  upon  the  center  in  an  irregular  and 
alternating  manner  which  apparently  depends  upon  the  ratio 
of  height  to  length  of  orifice,  as  indicated  in  the  following 
table  of  experimental  coefficients  for  square  edged  orifices  in 
thin  plate  and  with  full  contraction,  which  were  determined 
by  Poncelet  and  Lesbros. 

Coefficients  of  discharge  for  square  edged  orifices  in  thin 
plate  and  with  full  contraction. 


SULLIVAN'S  NEW  HYDRAULICS.  247 

TABLE  No.  42— Table  from  Ponoelot  and  Lesbros. 


Dimensions  of  Orifice  in  Inches. 

Head   above 

Center 

8X8 

6X8 

4X8 

3X8 

2X8 

1X8 

0.4X8 

In  Inches 

0.4 

.70 

0.8 

.65 

.69 

1.0 

.64 

.68 

1.5 

.61 

.64 

.68 

2.0 

.60 

.62 

.64 

.68 

2.5 

.59 

.61 

.62 

.64 

.67 

3.0 

.60 

.61 

.62 

.64 

.67 

3.5 

.57 

.60 

.61 

.62 

.64 

.66 

4.0 

m 

.58 

.60 

.61 

.63 

.64 

.66 

4.5 

.56 

.59 

.60 

.61 

.63 

.64 

.66 

5.0 

.57 

.59 

.61 

.62 

.63 

.64 

.66 

8.0 

.59 

.60 

.61 

.62 

.63 

.64 

.65 

12.0 

.60 

.60 

.61 

.62 

.63 

.63 

.64 

36.0 

.60 

.60 

.61 

.62 

.62 

.63 

.63 

60.0 

.60 

.60 

.61 

.61 

.62 

.62 

120.0 

.60 

.60 

.60 

.60 

'.60 

.61 

.61 

TABLE  No.  43 — Table  From  George  Rennie. 


Head   above 

Center 

Dimensions  of  Orifice. 

Coefficient 

In  Feet 

1.0 

1  inch  diameter,  circular. 

.633 

1.0 

1X1  inches,  square. 

617 

1.0 

2.0 

1  square  inch  area,  triangular. 
1  inch  diameter,  circular. 

!596 
.619 

2.0 

1X1  inch,  square. 

.635 

2.0 
3.0 

1  square  inch  area,  triangular. 
1  inch  diameter,  circular. 

.577 
.628 

3.0 

1X1  inch,  square. 

.606 

3.0 
4.0 

1  square  inch  area,  triangular. 
1  square  inch  area,  triangular. 

.572 
.5«3 

4.0 

1X1  inch,  square. 

.M 

SULLIVAN'S  NEW  HYDRAULICS. 


TABLE  No.  44— Table  from  Gen.  Ellis. 

Coefficients  of  discharge  for  square  edged.circular  Orifices 
in  iron  plate  one  half  inch  thick. 


Head  above 
Center 
In  Feet 

Diameter  of  Orifice  in  Feet. 

Coefficient 

2.1516 
9.0600 
17.2650 
1.1470 
10.8819 
17.7400 
1.7677 
5.8269 
9.6381 

0.50 
0.50 
0.50 
1.00 
1.00 

ft 

2.00 
2.00 

.60025 
.60191 
.59626 
.5T<73 
.59431 
.59994 
.58829 
.60915 
.61530 

TABLE  No.  45. 

Coefficients  for  square  orifice  1X1  foot  with  curved  en- 
trance and  discharge  slightly  submerged.    (Gen.  Ellis.) 


Head  above 
Center 
In   Feet 

Dimensions  of  Orifice. 

Coefficient 

3.0416 
10.5398 
18.2180 

Square,  1X1  Feet 
Square,  1X1  Feet 
Square,  1X1  Feet 

.95118 
.9*246 
.94364 

In  these  last  experiments  if  the  orifice  had  been  in  the 
'orm  of  the  vena  contracta,  and  the  discharge  had  been  en- 
tirely free  instead  of  being  partially  under  water,  it  is  proba- 
ble that  the  coefficient  would  have  reached  .98,  and  would 
not  have  been  affected  in  any  manner  except  by  the  slight  re- 
sistance of  efflux  offered  by  the  perimeter  of  the  orifice.  The 
curving  entrance  had  almost  suppressed  all  contraction  of 
the  jet  in  the  above  experiments. 

The  object  of  these  tables  and  suggestions  is  to  point  out 
the  fact  that  all  these  uncertainties  in  the  application  of  weir 
and  orifice  coefficients  may  be  easily  avoided  by  so  chamfer- 
ing the  inner  edges  of  the  weir  notch  or  orifice  as  to  make 
them  conform  as  nearly  as  possible  to  the  form  of  the  vena 
contracta. 

In  many  cases  of  the  practical  application  of  the  ordinary 


SULLIVAN'S  NEW  HYDRAULICS.  249 

weir  and  orifice  coefficients  the  conditions  are  such  that  com- 
plete contraction  cannot  be  obtained.  In  almost  any  case  it 
is  much  more  convenient  to  suppress  all  contraction  than  to 
obtain  complete  contraction,  and  when  contraction  is  sup- 
pressed there  is  no  limit  to  the  range  of  the  remaining  coeffi- 
cient which  should  be  determined  in  the  same  manner  as  the 
value  of  m  for  pipes  or  open  channels.  When  contraction  is 
suppressed  (as  pointed  out  §  83).  Then  for  a  submerged 
orifice. 

v  =A/§IZ£L=8.025 J3Z£1;   m-6^H 


q  =AreaX8.025j  H*/r>   = 

v       m 

And  for  weirs, 

28.6225  H  r/r« 
= — 


And. 


m 

When  the  numerical  value  of   m  is  ascertained  for  any 
thickness  of  plate  it  will  apply  to  any  shape  or  size  of  orifice  or 

weir  notch,  and  the  square  root  of  its  reciprocal^/ ,    may 

then  be  taken  and  combined  with  the  constant  8.025  or  5.35  as 
the  case  may  be. 


APPENDIX  II. 


Useful  Data  and  Tables  Relating  to   Water  Works  and  the 
Water  Supply  of  Cities  and  Towns. 

97.— Purposes  to  Which  City  Water  is  Applied. 

In  planning  a  water  works  system  for  town  or  city  sup- 
ply, the  nature  of  the  chief  occupation  of  the  inhabitants 
must  be  considered  as  well  as  the  number  of  inhabitants  at 


250  SULLIVAN'S  NEW  HYDRAULICS. 

present,  and  the  probable  increase  in  population  within  the 
next  fifteen  or  twenty  years.  The  purposes  to  which  city 
water  will  be  applied  will  depend  upon  the  humidity  of  the 
climate.  In  the  arid  portion  of  the  West  the  city  water  is 
demanded  for  all  purposes  to  which  water  is  applied,  such  as 
irrigation  of  lawns  and  gardens,  and  shade  trees,  street 
sprinkling,  carriage  washing,  watering  horses  and  cows, 
water  for  steam  boilers  and  hydraulic  motors,  hydraulic  lifts 
or  elevators,  steam  laundries,  drinking  fountains,  ornamental 
fountains,  manufacturing  purposes,  extinguishment  of  fires 
and  ordinary  household  uses.  Where  manufacturing  is  the 
chief  business  of  a  town  the  demand  for  water  will  be  two  or 
three  hundred  per  cent  greater  than  in  towns  of  equal  size  and 
in  like  climates  which  are  not  manufacturing  centers.  In 
some  manufacturing  towns  situated  on  rivers  the  factories 
have  their  own  private  water  supply,  and  in  such  cases  the 
city  water  works  is  called  upon  only  for  water  for  ordinary 
purposes.  The  coast  states,  and  the  Eastern  and  Southern 
states,  have  frequent  and  large  rainfalls  and  except  at  manu 
facturing  centers,  the  city  water  works  in  these  states  will 
not  be  called  upon  except  for  ordinary  purposes.  In  the  arid 
portion  of  the  West  the  demand  on  the  city  water  supply  is 
from  fifty  to  one  hundred  per  cent  greater  than  in  towns  of 
like  population  in  other  parts  of  the  United  States. 

In  non-manufacturing  towns  in  such  climates  as  in  Ar- 
kansas, Mississipi  and  Louis&na,  the  demand  for  all  purposes 
will  not  exceed  60  gallons  per  capita  per  24  hours,  while  in 
Colorado  and  other  arid  states  the  demand  in  small  non- 
manufacturing  towns  is  from  110  to  150  gallons  per  capita 
per  24  hours,  and  in  older  and  larger  cities  the  demand  is 
from  150  to  200  gallons  per  capita.  Should  an  essentially 
manufacturing  city  spring  up  in  the  arid  West,  it  is  probable 
that  the  demand  for  water  would  reach  400  gallons  per  capita 
per  24  hours. 

98.— Quantity  of   Water   per    Capita   Required— The 

quantity  of  water  required  per  capita  per  24  hours  for  the 
.present  given  number  of  inhabitants,  and  for  all  purposes, 


SULLIVAN'S  NEW  HYDRAULICS. 


251 


depends  upon  the  bection  of  the  country  and  the  chief  occu- 
pation of  the  inhabitants,  as  just  pointed  out.  But  in  plan- 
ning a  water  supply,  the  very  rapid  increase  in  the  popula- 
tion of  towns  and  cities  in  the  United  States  must  be  amply 
allowed  for.  The  U.  S.  census  of  1890  shows  that  our  popu- 
lation is  f  aet  gathering  into  the  towns  and  cities.  The  popu- 
lation of  towns  and  cities,  taken  collectively,  throughout  the 
United  States,  increa  sed  by  61.10  per  cent  from  1880  to  1890, 
while  the  total  population  of  town  and  country  increased 
only  24.85  per  cent.  The  following  table  is  valuable  in  this 
connection. 

TABLE  No.  46. 

Growth  of  population  in  cities  and  in  the  United  States. 


Cen- 
sus 
Year 

Total  Pop. 
U.S. 

Population 
in  Cities 

Increase 
in  total 
pop.    per 
cent 

Per  cent 
total    pop 
in  citie 

of    the 
living 
s. 

1800 

5,308,483 

210,873 

1810 

7,239,881 

356,920 

36.28 

4.93 

1820 

9,633,822 

475,135 

33.66 

4.93 

1830 

12,866020 

1,864509 

32.51 

6.72 

1840 
1850 

17,069,453 
23191,876 

1,453.994 
2,897,586 

32.52 
35.83 

8.52 
12.49 

1860 

31,443,321 

5,072,256 

35.11 

16.13 

1870 
1880 

38,558,371 
50,155,783 

8,071,875 
11,318,547 

22.65 
30.08 

20.93 
22.57 

1890 

62,622,250 

18,238,672 

24.85 

29.12 

99.— Table  Showing  the  Consumption  of  Water  Per 
Capita  Per  24  Hours  in  Various  Cities  and  Towns,  and 
the  Cost  to  the  Consumer  Per  1,000  Gallons,  and  the  in- 
crease In  Population  in  Each  City  in  20  Years. 

The  foregoing  table  shows  that  the  general  average  in- 
crease of  population  in  the  towns  and  cities  of  the  United 
States  was  61.10  per  cent  from  1880  to  1890.  But  the  rate  of 
increase  varies  in  different  sections  of  the  country  and 
also  in  different  classes  of  cities  and  towns.  The  railroad 
and  general  manufacturing  centers  increase  most  rapidly  in 
all  parts  of  the  country,  while  tho  general  growth  of  all  cli 
of  towns  increases  most  rapidly  in  the  Western  states. 


SULLIVAN'S  NEW  HYDRAULICS. 


TABLE  No.  47. 


a 

P 

cCt 

So 

1 

l-H 

1 

II 

11 

s 

• 

i 

• 

gg 

°  s 

a 

a 

00 

O 

<3 

6 

S 

o 

OS 

Alabama 

Birmingham 

0 

400 

26,241 

155 

8c  to  30  c 

Gala. 

Los  Angeles 

5,758 

11,183 

50,394 

175 

20c 

Colo. 

Denver 

4,759 

35,629 

106,670 

200 

Conn. 

New  Britain 

9840 

11,800 

19,010 

87 

10  c 

Conn. 
Conn. 

Norwich 
Hartford 

16,653 
37,180 

15.112 
42,015 

16,195 
53,182 

50 
125 

15  c  ;o  30  c 
7Hcto30c 

Conn. 

New  Haven 

50840 

62882 

85,981 

130 

fV4cto30o 

Georgia 
Georgia 

Atlanta 
Augusta 

21,789 
15,389 

37,409 
21,891 

65.515 
33,1'  2 

164 
106 

10  c 

Georgia 

Macon 

10,810 

12,479 

22,698 

70 

6  c  to  30  c 

Illinois 

Aurora 

11.162 

11,873 

19,634 

60 

Illinois 
Illinois 

as?80 

298,977 
5441 

503,185 

8,787 

1,098,576 
17,429 

131& 
70 

8  c  to  10  c 
3c  to8c 

Illinois 

Streator 

1,486 

5,157 

6.671 

120 

10  c  to25c 

Illinois 
Illinois 
Indiana 

Freeport 
LaSalle 
Indianopolis 

7,889 
5,200 
48,224 

8,516 

7,847 
75,056 

10,159 
11,610 
107,445 

46 
70 
90 

10  c  to  50  c 
8  c  to  \~>  c 
6  c  to  30  o 

Indiana 
Iowa 

Richmond 
Cedar   Rapid* 

9,445 
5,940 

12,472 
10,104 

16,845 
17,997 

74 
68 

5  c  to  25  c 
lOctoSOc 

Iowa 

Sioux  City 

3,401 

7,366 

37,862 

43 

10  c  to  25  c 

Iowa 

Des  Moines 

5,241 

22408 

50,067 

43 

20cto  40  o 

Kansas 

Atchison 

15,105 

14,122 

90 

10  c  to  50  c 

Kansas 

Minneapolis 

2,000* 

200 

35  c 

Kansas 

Arkansas  Citj 

3,347 

46 

lOcto  40c 

Ky. 

Louisville 

100,752 

123,758 

161,005 

JO 

6  c  to  15  c 

Ky. 

Lexington 

14.801 

16,656 

22,355 

40 

10  c  to  25  c 

Ky. 

Frankfort 

5,396 

6.958 

8,500 

100 

6  c  to  15  c 

Ky. 

Fulton 

4,500* 

200 

Md. 

Hagerstown 

5.779 

6,627 

11,698 

115 

8  c  to  40c 

Mass. 

Adams 

12,090 

5,591 

9,206 

84 

Mass. 

Fall   River 

26.766 

48,961 

74,351 

28 

Mass. 

Holyoke 

10,733 

21,915 

35,528 

78 

5  c  to  15  c 

Mass. 
S  ass. 

Lowell 
New  Bedford 

40,928 
21,320 

59,475 
26845 

77,605 
40,705 

75 
113 

2  1/,  ctolSc 

ass. 

Newton 

12,825 

16,995 

24,357 

53 

12cto35o 

Mass. 
Mich. 

Springfield 
Battle  Creek 

26,703 

5,838 

33,340 
7,063 

44,164 
13,190 

87 
31 

30c 

Mich. 

Bay  City 

7,064 

20,693 

27,836 

80 

5  c  to  10  c 

Mich. 

Detroit 

79,577 

116,340 

205,669 

140 

3%c 

Mich. 

Miss. 

Sagnaw 
Vicksburg 

7,460 
12,443 

10,525 
11,814 

46,215 
13,298 

100 
43 

6  c  to  11  c 
6  c  to  35  c 

Missouri 

5,555 

6.522 

21.842 

SO 

25  c 

Missouri 

sfEolis 

310,864 

350,518 

460,357 

75 

10  c  to  30  c 

SULLIVAN'S  NEW  HYDRAULICS. 
TABLE  No.  47  CONTINUED. 


253 


!i 

|l 

S 

0 

ot 

OQ 

00 

00 

CO 

(2  « 

O§ 

£ 

p 
§      ' 

i 

a 

at 

P 

0) 

1> 

s* 

OQ 

0 

6 

Q 

0 

i* 

Missouri 
N.  Hamp 

Butler 
Nashua 

10,543 

13  397 

4,000* 
19,266 

25 
150 

6cto60c 
10  c  to  20  o 

N.  Hamp 

Manchester 

23,536 

32,630 

43,983 

50 

20  c 

N.'  Y. 

Bayonne 
Portland 

3.834 
3066 

9.372 
4050 

18,996 
8,561 

7? 
90 

13  &  c-23*c 

10  c 

N  .  Y. 

Elmira 

15863 

20541 

28.070 

86 

1%  c  to  45  o 

N  .  Y. 

Kingston 

6,315 

18,344 

21,181 

80 

8cto30c 

N.  Y. 

Olean 

7,3S8 

75 

10  c  to  40  c 

N.  Y, 

Syracuse 

43051 

51.792 

87.877 

300 

6  c  to  25  c 

N.  Y. 
N.  Y. 
N.  Y. 

Brooklyn 
\ewYorkCity 

396,099 
12.733 

942,292 

566,663 

18,892 
1,206,299 

804,377 
31.942 
1,513.501 

100 
70 
92 

IVt  C-11&  o 
4  c  to  25  c 

Ohio 

Dayton 

30,473 

33678 

58,868 

53 

*c 

Ohio 

Findlay 

3,315 

4.633 

18,672 

48 

6  c  to  12  c 

Ohio 

Oberlin 
Sanduskj 

13,000 

15,838 

4,000 
19,234 

20 
154 

30  c 
4  c  to  15  c 

Ohio 
Ohio 
Ohio 

Springfield 
Toledo 
Cincinnati 

12,652 
31,584 
216.239 

20,730 
50.137 
255,139 

32,135 
82,652 
296.309 

90 
70 
124 

10  c  to  40  c 
SotolOc 
17  c 

Oregon 
Penn. 

Salem 
Ml  City 

2276 

7,315 

10,943 

100 
230 

15  c  to  25  c 
6c  to25c 

Penn. 
Penn. 

tfcKeesport 
Williamsport 

2,523 
16,030 

8,212 
18,934 

20,711 
27,107 

110 

200 

4!4cto30c 
SctolOc 

Penn. 

^larrisburg 

23,104 

30,762 

40,164 

130 

2!4  c  to  lOc 

Penn. 

Philadelphia 

674  022 

847.170 

1,046.252 

143 

\o 

R  I. 

iVoonsocket 

11527 

16,050 

20,759 

22  y* 

10  c  to  80  c 

Texas 

Pawtucket 
Laredo 

6,619 
2,046 

19,0  0 
3,321 

2>,  502 
11,313 

79 
150 

tic.  to  30  c 
60  c 

Texas 

?ort  Worth 

0 

6,663 

20.725         130 

20  c  to  65  o 

Va. 

Richmond 

51.038 

66,600 

80,388     '    151 

7  c  to  15  o 

*Estimated  Population. 

The  above  table  will  be  useful  in  determining  the  quan- 
tity of  water  required  per  24  hours  per  person,  and  in  de- 
termining what  extia  capacity  of  reservoirs  and  conduits 
should  be  provided  for  the  increase  in  population  during  the 
coming  20  years.  The  capacity  of  a  water  supply  system 
should  not  be  based  on  the  present  number  of  inhabitants, 
but  upon  the  probable  number  of  inhabitants  20  years  hence. 


254  SULLIVAN'S  NEW  HYDRAULICS. 

What  the  increase  of  population  will  be  in  any  given  town  or 
city  within  any  given  number  of  years  is  a  matter  which 
must  be  considered  in  the  light  of  the  local  conditions  and 
surroundings  of  each  given  town  or  city.  There  are  very 
few  cities  or  towns  in  the  United  States  which  do  not  increase 
by  50  per  cent  within  20  years,  and  some  increase  by  from  300 
to  600  per  cent  within  ten  years.  The  general  average  increase 
of  population  in  all  cities  and  towns  in  the  United  States  for 
the  10  years,  1880-1890,  was  61.10  per  cent. 

100.~Formulas  and  Tables  for  Determining  the  Diam- 
eter of  the  Conduit  or  Pipe  Required  to  Convey  any 
Given  Number  of  Gallons  Per  24  Hours.  -When  the  total 
supply  of  water  in  gallons  per  24  hours  has  been  decided 
upon,  then  the  required  diameter  in  feet  of  the  circular  brick 
conduit  or  pipe,  or  other  circular  water  way,  may  be  at  once 
found  by  the  formula 


In  this  formula  the  value  of  m  varies  with  the  class  or 
roughness  of  the  internal  circumference  of  the  waterway, 
and  the  value  of  m  must  be  in  terms  of  diameter  in  feet. 
The  value  of  m  for  any  class  of  wet  perimeter  will  be  found 
by  referring  to  the  different  groups  of  pipes  and  channels.  If 
the  value  of  m,  when  found,  is  in  terms  of  R  in  feet,  it  may 
be  converted  to  terms  of  d  in  teet  as  shown  at  section  10. 

In  the  above  formula  q=cubic  feet   per    second,  and   S 

TT 

=the  sine  of  the  inclination  of  the  waterway=  -j-. 

For  a  constant    degree   of   roughness    of    perimeter,  the 

11  /m2 
value  of  ~yf  ~^^  is  a    constant,  and    the    formula    may  be 

simplified  accordingly.     Thus,  if  we    are    going    to    adopt  a 


SULLIVAN'S  NEW  HYDRAULICS.  255 

double  riveted  asphaltum  coated  Bteel   pipe,  then   m=.00033, 

11  /m8 
and  j/    ogQg  =0.2541,  and  the  formula  for   any   pipe  in   this 

11  /  Q4 
class   reduces  to  d=.2541-|// -g^.    If  we  adopt  an   ordinary 

uncoated,  cast  iron  pipe,  then  m=.0004,  and    the  formula  re- 
duces to 


d=  l 

If  the  pipe  is  to  convey  water  from  the  distributing  res- 
ervoir to  the  street  mains,  its  capacity  or  diameter  should  be 
such  as  to  enable  the  pipe  to  maintain  a  given  pressure  in  Ibs. 
per  square  inch  at  the  point  of  juncture  with  the  street  sys- 
tem while  it  is  supplying  the  given  quantity  of  water  in  cubic 
feet  per  second.  It  is  also  well  to  remember  that  the  total 
supply  per  24  hours  is  usually  drawn  between  6  o'clock  a.  m 
and  9  o'clock  p.  m.,  and  for  this  reason  the  city  supply  pipe 
leading  from  the  distributing  reservoir  to  the  city  must  have 
such  diameter  as  will  pass  the  entire  24  hours'  supply  within 
12  hours'  time,  and  also  maintain  a  given  pressure  while  so 
discharging.  In  other  words  this  pipe  must  carry  a  given 
quantity  of  water  within  a  given  time  with  a  given  loss  of 
pressure  or  head  at  a  given  point.  The  formula  for  finding 
the  required  diameter  to  carry  a  given  quantity  with  a  given 
or  pre-determined  loss  of  head  has  already  been  given  and 
discussed .  (See  §  §  64, 63.)  The  following  tables  will  greatly 
facilitate  all  such  calculations,  and  show  at  once  the  value  of 
q,  or  cubic  feet  per  second,  corresponding  to  any  supply  in 
gallons  per  24  hours.  (See  also  §  102) . 


256  SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  No.  48.* 


Gallons   pei 
24  hours= 

Cub  feet 
per  sec- 
ond q 

Loga 
rithm 
of  q 

Value  of  q4 

l. 

.00000154667 

6.189397 

.000,000,000,000,000,000,0  0,005,722,563,8T)0,675 

10. 

.0000154667 

5.189397 

.000,000,000,000,000,000,057,225,638,506,75 

100. 

.000154667 

4.189397 

.  030,000,000,000,0  X),572,256,335.067  .5 

1,000. 

.00154667 

3.189397 

.000,000,000,005,722,563,850,675 

10,000. 

.0151667 

2.189397 

000,000,057,225,638,506,75 

100,000 

.154667 

1.189397 

.000,572,256,335,067,5 

1,000,000. 

1.54667 

0.189397 

5.722,568,850,675 

10,000,000. 

15.4667 

1.189297 

57,225.638,506,75 

100,000,000. 

ir,4.667 

2.189397 

572,256,385.067,5 

1,000,000,000. 

1546.67 

1.  189397 

5,722,563,850,675.034,2 

10.000,000,000. 

15466.70 

4.189397 

57,225,638,506,750,342.032,1 

100,000,000000. 

154667.00 

5.189397 

572,256,385,067,503,420,321  .00 

*One     cubic     foot=7.48     gallons.     One     gallon=23l    cubic 
inches. 

TABLE  No.  49. 


Gal.  per 
24  hours 

Cub.  feet 
per  sec- 
ond q 

Loga- 
rithm   of 

q 

Value  of  q4 

10 

.0000154667 

5.1893r8 

.000,000,000,000,000,000,057,225,638,508.75 

20 

.0000309334 

5.490427 

.000,000,000,000,000,000,915,720,15 

25 

.00003866675 

5.587337 

.000,000,000,000,000,002,235,678,513 

30 

.0000464001 

5.666518 

.000,000.000,000,000,004,63) 

40 

.0000618668 

5.791458 

.  000,000,000,000,000,014,65 

50 

.0(00773835 

5.888367 

.  000,000,000,000,000,035,77 

60 

.0000928002 

5.967549 

.  000,000,000,000,000,074,26 

SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  No.  49— Continued. 


257 


Gal.  per 
24  houre 

Cub.  feet 
per  sec- 
ond q 

Loga- 
rithm  of 

q 

Value  of  q*. 

70 

.00  1082669 

4.034495 

.000,000,000,000,000.137,398,72 

80 

.0001237336 

4.092488 

.000,000,000,000,000,234,396,1 

90 

.0001392003 

4.1436399 

.000,000,000,000,000,375,457,278 

100 

.000154667 

4.189398 

.000,000,000,000,000.572,^56,385,067.5 

200 

.000309334 

4.490427 

.000,000,000,000,009,157,201,5 

250 

.0003866675 

4.587337 

.000,000,000,000,022,356,735.513 

300 

.000464001 

4.666518 

.  000,000,000,000,046,350 

400 

.000618668 

4.791458 

.  000,000,000,000,146,500 

500 

.000773335 

4.888367 

.  000,000,000,000,357,700 

600 

.000928002 

4.967549 

.  000,000,000,000,742,600 

700 

.001082669 

3.034496 

.000,000,000,001,373,987,200 

800 

.001237336 

3.092488 

.000,OOU,000,002,343,96l,00 

900 

.001392003 

8.1436399 

.000,000,000,003,754,572,780 

1,000 

.00154667 

3.189398 

.000,000,000,005,722,563,850,675 

2,000 

.00309334 

3.493427 

.000,000,000,091,572,015 

2,500 

.003866675 

3.587337 

.000,000,000,223,-567,351,3 

3,000 

.00464001 

3.666518 

.000,000,000,463,5 

4,000 

.00618668 

3.791458 

.000,000,001,465 

5,000 

.00773335 

3.888367 

.000,000,003,577 

6,000 

.00928002 

3.967549 

.000,000,007,426 

7,000 

.01082669 

2.034495 

.000,000,013,739,872 

8,000 

.01237336 

2.092488 

.000,000,023,439,610 

9,000 

.01392003 

2.5745399 

.000,000,037,545,727,800 

10,000 

.0154667 

2.189398 

.000,000,057,225,638,506,750 

20,000 

.0309334        J2.490427 

.000,000,915,720,015 

25.000 

.03866675 

2.587337 

.000,002,223,567,351,3 

30,000 

.0464001 

2.666518 

.000,004,635 

258  SULLIVAN'S  NEW  HYDRAULICS 

TABLE  No.  49— Continued. 


Gal.  per 
24  hours 

Cub.  feet 
per  sec- 
ond q 

Loga- 
rithm   of 

q 

Value  of  q* 

40,000 

.0618668 

2.791458 

.000,014,650 

50,000 

.0773335 

2.888367 

.000,035,770 

60.000 

.0928002 

2.967549 

.000,074,260 

70,000 

.1082669 

1.034495 

.000,137,398,720 

80,000 

.1237336 

1.092488 

.000,234,396,100 

90,000 

.1392003 

1.5745399 

.000,375,457,278 

100000 

.154667 

1.189398 

.000,572,256,385,067,5 

200,000 

.309334 

1.490427 

.(09,157,201,5 

250,000 

.3866675 

1.587337 

.022,356,735,130 

300,000 

.464001 

1.666518 

.046,35 

400,000 

.618668 

1.791458 

.146,5 

500,001 

.773335 

1.888367 

.357,7 

600,000 

.928002 

1.967549 

.742,6 

700000 

1.082669 

0.034495 

1.373,987,2 

800,000 

1.237336 

0.092488 

2.343,961 

900,000 

1.392003 

0.1436399 

3.754.572,78 

1,000,000 

1.54667 

0.189398 

5.722,563,850,675 

2,000,000       3.09334 

0.490427 

91.572,015 

2,500,000 

3.866675 

0.587337 

223.567,351,3 

3,000,000 

4.64001 

0-666518 

463.500 

4,000,000 

6.18668 

0.791458 

1465.00 

5,000000 

7.73335 

0.888367 

3577.00 

6,000,000 

9.28002 

0.967549 

7426.00 

7,000,000 

10.82669 

1.034495 

13939.872 

8,000.000 

12.37336 

1.092488 

23439.61 

9,000,000 

13.92003 

1.1436399 

37545.727,8 

SULLIVAN'S  NEW  HYDRAULICS. 

TABLE  49— Continued. 


259 


Gal.  per 
24  hours 

Cub.  feet 
per  sec- 
ond q 

Loga- 
rithm  of 

q 

Value  of  q*. 

10.000,000 

15.4667 

1.189398 

57225.638,506,75 

20,000,000 

30.9334 

1.490427 

915720.16 

25,000,000 

38.66675 

1.587337 

2235673.513,03 

30,000,000 

46.4001 

1.666518 

46350'  0.00 

40,000000 

61.8668 

1.791458 

14650000.00 

50,000,000 

77.3335 

1.888367 

35770000.00 

60,000,000 

92.8002 

1.967549 

74260000.00 

70000,000 

108.2669 

2.034495 

137398720.00 

80,000,000 

123.7336 

2.092488 

234396100.00 

90.000,000 

139.2003 

2.1436399 

375457278.00 

100000,000 

154.667 

2.189398 

572256385.067,5 

200,000,000 

309.334 

2.490427 

9157201500.000 

250,000,000 

386.6675 

2.587337 

22356735130.30 

101.—  To  Find  the  Diameter  in  Feet  of  a  Circular  Con- 
duit or  Pipe  With  Free  Discharge,  as  From  One  Reservoir 
Into  Another,  which  is  required  to  Discharge  a  given  quan- 
tity in  Cubic  Feet  Per  Second,  the  Total  Head  or  ths  Slope 
Being  Known:  — 

The  general  formula  for  finding  the  required  diameter  in 
feet  will  be 


Simplifying  the  formula  as  pointed  out  heretofore  (§100) 
and  for  the  following  classes  or  degrees  of  roughness  of  peri- 
meter we  have, 

(1)  For  ancoated  clean  cast    iron    pipe,m=.0004,  and 


SULLIVAN'S  NEW  HYDRAULICS. 


(2)  For  uncoated  steel  or  wrought  iron,  m=.00038,  and 


(3)  For  uncoated  wooden    stave  pipe,    made  of    dressed 
hard  wood,  and  closely  jointed,  m=.00048,  and 


(4)  For  cement  mortar  lined  pipe,  one  third  sand  and  two 
thirds  cement,  m-.  000424,  and 


(5)  For  riveted  pipe,  thoroughly  dipped  and   coated   with 
asphaltum  and  crude  petroleum,  m=.  000325,  and 


(6)  For  cast  iron  or  welded   pipes   thoroughly   asphaltum 
coated  and  carefully  laid  and  jointed,  m=r.000305,  and 


For  brick  perimeters  see  §  24.  Always  make  an  extra  al- 
lowance in  the  diameter  of  pip  >  or  conduit  for  future  deteri- 
oration and  for  deposits. 

102.—  To  Find  the  Diameter  in  Feet  of  a  Circular  Con- 
duit or  pipe  which  is  Required  to  carry  a  Given  Quantity  in 
Cubic  Feet  Per  Second  to  a  Given  Point  and  Maintain  a 
Given  Head  or  Pressure  at  That  Point  while  Delivering  the 
Required  Quantity:  — 

This  formula  is  very  important  in  designing  power  mains 
for  water  wheels.'in  which  it  is  required  to  maintain  a  given 
pressure  or  head  at  the  base  of  the  nozzle  which  discharges 
upon  the  wheel  or  motor  It  applies  equally  well  to  hydraulic 
giants  used  in  placer  mining,  and  to  fire  hose  with  nozzle  at- 
tached, and  to  all  other  cases  where  the  discharge  is  partially 


SULLIVAN'S  NEW  HYDRAULICS.  26l 

throttled,  as  in  the  case  of  a  supply  pipe  leading  from  the 
distributing  reservoir  of  a  water  works  system  to  the  street 
mains.  In  the  latter  case  it  is  desirable  to  so  proportion  the 
diameter  that  it  will  convey  the  required  quantity  of  water 
and  at  the  some  time  maintain  not  less  than  a  given  head 
pressure  at  the  point  of  its  juncture  with  the  street  mains. 
The  general  formula  will  be 


In  which, 

h"  =  total  head  in  feet  to  be  lost  in  friction  in  the  length  I. 

n=coefficient  of  resistance,  and  varies  with  different 
classes  of  wet  perimeter. 

Simplifying  the  formula  for  given  classes  of  perimeters 
as  heretofore  pointed  out  (§§  64,  68)  and 

(1)    For  uncoated  clean  cast  iron,  n=.0003938.  and 


(2)    For  uncoated  clean  steel  orwrought  iron,  n=.00037411 
and 


(3)    For  uncoated  wooden  stave   pipe,   made  of  dressed 
hard  wood  and  closely  jointed,  n=.00047256,  and 


(4)    For  cement  mortar  lined   pipe,  one-third  sand   and 
two-thirds  cement,  n=.0004175,  and 

d=  .265311 


262  SULLIVAN'S  NEW  HYDRAULICS. 

(5)    For  riveted  pipe,  thoroughly  dipped  and  coated  with 
asphaltum  and  crude  petroleum,  n=.00032,  and 


(6)  For  cast  iron  and  welded  pipes,  thoroughly  coated 
with  asphaltum  and  oil,  and  carefully  laid  and  jointed,  n= 
.00030,  and 


REMAKK. — For  any  given  class  of  perimeter  n=mX-9845, 
and  m=-qgT^,  and  the  difference  in  value  between 

H/'  m*          11  /~~n*~ 
y     ogQc  and  ./  •qgnc  for  any  given  roughness  is  equal  .0008. 

That  is  to  say, 

y1    OQQ5    is  .0008  less  than  the   corresponding  ualue  of 

n/  m» 
^   ^805- 

11  /  m*~  11  /~n*~~ 

If    ,/  -oon^- =-2608, then  1//^?^=.2600,  and  soon. 

V      .OoUD  V      .ooUO 

While  the  difference  in  value  of  m  and  n  is  small,  yet  it 
must  be  remembered  that  m=the  head  per  foot  length  of 
pipe  to  balance  the  resistance  and  generate  the  mean  velocity 
of  flow,  and  n  is  equal  the  friction  head  only,  per  foot  length 
of  pipe.  In  a  pipe  of  considerable  length  the  difference  be- 
comes very  considerable.  (See,  in  this  connection,  §§  4  and  5) 

Formulae  (43)  and  (45)  given  in  §17  may  be  adopted  in- 
stead of  the  above  but  in  that  event  the  value  of  m  or  n  must 
be  converted  to  terms  of  P  as  in  §  17. 


SULLIVAN'S  NEW  HYDRAULICS.  263 

103—  Velocities,     Discharge     and   Friction    Heads    for 
Slopes  and  Diameters. 

The  slope  required  to  generate  a  velocity  of  one   foot  per 
second  in  any  given  diameter  with  full  and  free  discharge  is 


The  slope  required  to  generate  any  other  velocity,   either 
greater  or  less  than  one  foot  per  second,  is 


In  the  latter  formula  v»  must  equal  the  square  of  the  de- 
aired  velocity  in  feet  per  second.  Having  found  the  value  of 
S  for  v^l.OO  in  any  given  diameter,  then  the  required  value 
of  S  to  generate  any  other  velocity  in  the  given  diameter, 
will  equal  the  value  of  S  for  v  =  1.00  multiplied  by  the  square 
of  the  proposed  velocity  .  The  distance  or  length  in  feet  i,  of 
pipe,  in  which  there  is  a  fall  of  one  foot  is 


As  the  value  of  S  shows  the  total  head  per  foot  length  of 
pipe,  the  fall  in  feet  per  100  feet  length  is  found  by  moving 
the  decimal  point  in  the  value  of  S  two  places  to  the  right. 
The  friction  head  per  foot  length  in  any  given  uniform  diam- 
eter with  full  and  free  discharge  is  .9845  per  cent  of  the  value 
of  S  for  that  pipe.  The  friction  head  may  therefore  be  easily 
found  from  the  value  of  S.  The  friction  head  per  100  feet 
length  of  pipe  will  be 

h*=SX98.45,  or  h  '  =(SX  100)—  (SX100X  -0155). 

When  the  friction  head  per  100  feet  is  ascertained  for  a 
given  diameter  with  v=1.00,  then  the  friction  head  per  100 
feet  in  the  given  diameter  for  any  other  velocity  will  equal 
that  for  v=1.00  multiplied  by  the  square  of  the  proposed  vel- 
ocity. The  following  table  (No.  50)  is  based  on  m=.0004  for 
all  clean  iron  pipes. 


264 


SULLIVAN'S  NEW  HYDRAULICS. 


TABLE  No.  50. 

Velocities,  Discharge  and  friction  heads  for  given  slopes 
and  diameters. 


d;/d 


g 

8 

h 

u 

06 

* 

a& 

8,1 

If 

?! 

h 

50 

•si^ 

>fc  0 

||| 

U| 
all 

Dischar 
Cubic  t 
Second. 

3  inches 

.25  feet 

0.500 

.0008 

0.50 

.07876 

11.01804 

.02455 

.0032 

1.00 

.31504 

22.03608 

.04910 

.0072 

1.50 

.70*84 

33.05412 

.07365 

.0128 

2.00 

1.26016 

44.07216 

.09820 

.0200 

2.50 

1.96900 

55.09020 

.12275 

.0288 

3.00 

2.83536 

66.10824 

.14730 

.0392 

3.50 

3.85928 

77.12628 

.17185 

.0512 

4.00 

5.04070 

88  14432 

.19640 

.0648 

4.50 

6.37964 

99.16236 

.22095 

.0800 

.    5.00 

7.87600 

110.18040 

.24550 

.0968 

5.50 

9.53008 

121.19844 

.27005 

.1152 

6.00 

11.34158 

132.21648 

.29460 

.1352 

6.50 

13.31060 

143.23452 

.31915 

.1568 

7.00 

15.43715 

154.25256 

.34370 

.2048 

8.00 

20.16281 

176.28864 

.39280 

.3200 

10.00 

31.50400 

220.36080 

.49100 

4  inches 

.3333  feet 

0.579 

.00207850 

1.00 

.204625 

39.18024 

.08730 

.00466662 

1.50 

.459428 

58.77036 

.13095 

.0083140 

2.00 

.8185133 

78.36048 

.17460 

.0129906 

2.50 

1.278924 

97.95060 

.21825 

.0187065 

3.00 

1.811655 

117.54080 

.26190 

.0254616 

3.50 

2.506694 

136.72096 

.30555 

.0332560 

4.00 

3.254053 

155.90120 

.34920 

.0420896 

4.50 

4.143721 

175.08144 

.392S5 

.0519625 

5.00 

5.115708 

194.26168 

.43650 

.0628746 

5.50 

6.190004 

213.44192 

.48015 

.0748260 

6.00 

7.368619 

232.62216 

.52380 

6.50        8.645544 

251.80240 

.56745 

.'1018465 

7.00       10.026788 

270.98264 

.61110 

SULLIVAN'S  NEW  HYDRAULICS. 
TABLE    50.— Continued. 


o 
® 

8 

u 

^ 

!* 

02 

g| 

Is* 

lo-S 

§£ 

^^ 

O> 

O  *** 

v>  *^  ^ 

•g.2  o 

J' 

.s 

?! 

002 
02 

?!1 

ill 

.1-5.2 

fiu  S 

5  inches 

4167  feet 

0.645 

.00148120 

1.00 

.14651329 

61.17144 

.13630 

.00334845 

1.50 

.32965490 

91.757160 

.20445 

:()05952<SO 

2.00 

.58605316 

122.342880 

.27260 

.00930125 

2.50 

.91570806 

152.92860 

.34075 

!  "1339380 

3.00 

1.31861961 

183.51432 

.40890 

.01823045 

8.50 

1.79276780 

214.10004 

.47705 

.02381120 

4.00 

2.34421264 

244.68576 

.54520 

.03013605 

.03720500 

4.50 
5.00 

2.96989412 
3.66283225 

275.27148 
305.85720 

.61335 
.68150 

.04501805 

5.50 

4.43202702 

336.44292 

.74965 

.05357520 

6.00 

5.27447844 

367.02864 

.81780 

6  inches 

.50  foot 

0.7071 

.001131156 

1  00 

.1113623 

88.14432 

.1964 

.002543101 

1  50 

.2503683 

132.21648 

.2946 

.004524624 
.007%9725 

2.00 
2  50 

.4454492 
.6960144 

176.28864 

.3928 
.4910 

.010180404 

3  00 

1  002260 

264  [43296 

.5892 

.013*56051 
.DIM  19*49(5 
.022905909 
.028278900 

3.50 
4.00 
4.50 
5.00 

1.364198 
1.781796 
2.255086 
2.784057 

308.50512 
352.57728 
396.64944 
440.72160 

.6874 
.7856 
8838 
.9820 

.03J217469 

5  50 

3.368709 

484.79376 

[.0802 

.040721616 

6  00 

4.008903 

5?8.  86592 

[  1784 

7  inches 

.047791341 

6.50 

4.705057 

572.93808 

[.2766 

.5833  feet 

0.764 

.000897583 

1  '0 

.088367 

119.96424 

.26730 

.002019561 
.003590332 

1.50 
200 

.198825 
.&5346S 

179.94636 

239.92J4S 

.40095 
.53460 

.Ol)560lHt3 

2  50 

.552294 

299  91060 

.66825 

.00-078247 

3.00 

795303 

359.89272 

.80190 

.010995391 
.013961328 
.018176055 

3.50 
4.00 
4.50 

1.081996 
1.374493 
1.789433 

419.87484 
479.85(596 
539.83908 

.93555 
1.06920 

1.20285 

8  inches 

.022439575 
027151885 

5^50 

2.209176 
2.673103 

599.82120 
659.80332 

1.33650 
1.47015 

.6667  feet 

0.817 

.000734357 

1  00 

.0722975 

156.67608 

.3491 

.002937428 
.006609213 
.011749712 
.018358925 
.026436852 
.035983493 
.046998848 

2:00 

3.00 
4.00 
5.00 
6.00 
7.00 
8  00 

!  287  1888 
.6506771 
1.1567592 
1.8074362 
2.6027091 
3  5425749 
4  6270366 

313.35216 

470.02824 
626.70432 
783.38040 
940.05648 
1096.73256 

.6982 
1.0473 
1.3964 
1.7455 
2.0946 
2.4437 
2.7928 

.059482917 
.073435700 

9  00 
10.00 

5.8560932 
7.2297500 

li"li>  !(»472 
1566.76080 

3.1419 
3.4910 

SULLIVAN'S  NEW  HYDRAULICS 
TABLE  No.  50  CONTINUED. 


1 

as  "to 
* 

O    Jo 

<D  "S 

M 

1 

fc 

a 

-PH       O 

P 

lit 

fcfcS 

V 

I33 

££ 

li 

«3i 

S3l 

0    inches 
.75  Foot 

0.866 

.00061585835 

1.00 

.060631255 

198.27984 

.4418 

.00246343340 

2.00 

.242525020 

396.55968 

.8836 

.00554272415 

3.00 

.545681193 

594.83952 

1.3254 

.<HMS53733tiO 

4.00 

.970100080 

793.11936 

1.7672 

.01529f.45.s75 

5.00 

1.505936364 

991.39920 

2.2090 

.02217090060 

6.00 

2.182725170 

1189.67904 

2.6508 

.030177U:>915 

7.00 

2.970931474 

1387.95888 

3.0926 

.03941493440 

8.00 

3.880400300 

1586.23872 

3.5344 

.0498S452635 

9.00 

4.911131620 

1784.51856 

3.9762 

.06158583500 

10.00 

6.063125500 

1982.79840 

4.4180 

.07451886035 

11.  fO 

7.336381802 

2181.07824 

4.8598 

.08868360240 

12.00 

8.730900660 

2379.35808 

5.3016 

10  inches 

.8333    Ft 

0.913 

.00052576237 

1.00 

.051761306 

244.77552 

.5454 

.00210304948 

2.00 

.207045222 

489.55104 

1.0908 

.00473186133 

3.00 

465851748 

734.32656 

1.6  62 

.00841219792 

4.00 

.828180886 

979.10208 

2.1816 

.01314405925 

5.00 

1.294  32634 

1223.87760 

2.7270 

.01892744532 

6  00 

i.,s'->:uor,992 

1468.65312 

3.2724 

.02576235613 

7.00 

2.5363039H1 

1713.42864 

3.8178 

.03364879168 

8.00 

3.312723541 

1958.20416 

4.3632 

.04258675197 

9.00 

4.192665732 

2202.97968 

4.9086 

.05257623700 

10.00 

5.176130600 

2447.75520 

5.4540 

.06361724677 

11.00 

6.263117945 

2692.53072 

5.9994 

12  inches 

1.00  Feot. 

1.00 

.00040000000 

1.00 

03938 

352.48752 

.7854 

.0016 

2.00 

;  15752 

704.97504 

1.5708 

.0036 

3.00 

.35442 

1057.46256 

2.3562 

.0064 

4.00 

.63008 

1409.95(X'8 

3.1416 

.0100 

5.00 

.98450 

1762.43760 

3.9270 

.0144 

6.00 

1.41768 

2114.92512 

4.7124 

.0196 

7.00 

1.92962 

2467.41264 

5.4978 

.0256 

8.00 

2.52032 

2819.90016 

6.2832 

.0324 

9.00 

3.18978 

3172!  38768 

7.0686 

.0400 

10.00 

3.94800 

3524.87520 

7.8540 

.0484 

11.00 

4.76498 

3877.36272 

8.6394 

14  inches 

1.167  Feet 

1.080 

.00031736964 

1.00 

.026817735 

479.7672 

1.069 

00126947856 
.00285632676 

2.00 
3.00 

.1249*0165 
.281205370 

959.5344 
1439.3016 

2.138 
3.207 

.00507791424 

4.00 

.49!  1920,558 

1919.UC.NS 

4.276 

.00793424100 

5.00 

.78U26000 

2398.8360 

5.345 

.01142o30704 

6.00 

1.1248208115 

2878.6032 

6.414 

.01555111236 

7.00 

1.531007012 

3358.3704 

7.483 

.02031165696 

8.00 

1.999682628 

3838.1376 

8.552 

.02570694084 

9.00 

2.569280203 

4317.9048 

9.621 

.TO!  73696400 

10.00 

3.124504100 

4797.6720 

10.690 

.03840172844 

11.00 

3.790649969 

5277.4892 

11.759 

SULLIVAN'S  NEW  HYDRAULICS. 
TABLE  50— Continued. 


267 


o 

1' 

m 

? 

||. 

&! 

I4 

Is 
S-s 

?! 

Jh. 

02 

Ill 

111 

C  Q  m 

fcWfe 

11! 

Q  OS 

ill 

QOra 

16  Inches 

1.333  feet 

1.155 

00025980521 

1.00 

.025577823 

626.5248 

1.396 

00103922084 

2.00 

.103350513 

1253.0496 

2.792 

(X  1233,^24(^9 

3.00 

.23020  407 

1879.5744 

4.188 

0  '415688336 

4.00 

.409245167 

2506.0992 

5.584 

00649513025 

5.00 

.639445574 

3132.6240 

6.980 

00935298756 

6.00 

.920801626 

3759.1488 

8.376 

01273045529 

7.00 

1.253313324 

4385.6736 

9.772 

01662753344 

8.10 

1.636980668 

5012.1984 

11.168 

02104422201 

9.00 

1.971803657 

5638.7232 

12.564 

02598052100 

10.00 

2.5577823 

6265.2480 

13.960 

03143643041 

11.00 

3.126353005 

6891.7728 

15.356 

18  inches 

1.50   feet 

1.224 

0002124183 

1.00 

.02091259 

793.0296 

1.767 

0008496732 

2.00 

.08365033 

1588.0592 

3.534 

0019117737 

3.00 

.18821413 

2379  0888 

5.301 

0'>339;?6928 

4.00 

.33460131 

3172.1184 

7.068 

0053104575 

5.00 

.52281455 

3965.1480 

8.835 

0076470588 

6.00 

.75284194 

4758.1776 

10.602 

01'  140-4967 

7.00 

1.02471651 

5551.2072 

12.369 

0135947712 

8.00 

1.33840523 

6344.2368 

14.136 

0172"5S823 

9  00 

1.69391912 

7137.2664 

15.903 

0212418300 

10.00 

2.0912590 

7930.2960 

17.670 

0257026143 

11.00 

2.53  42238 

8723.3256 

19.437 

20  inches 

1.667  feet 

1.291 

000185865228 

1  00 

.0182984317 

979.2816 

2.182 

000743460912 

2.00 

.0731937268 

1958.5632 

4.364 

001672787052 

3  00 

.1646858853 

2937.8448 

6.546 

002973843648 
004646630700 

4.00 
5.00 

.2927749072 
.4574610000 

3917.1264 
4896.  40SO 

8.728 
10.910 

(X)tit.i9114.>20> 
009107396172 

6.00 
7  01 

.6587435411 

>9(«231532 

5875.6896 
6854.9712 

13.092 
15.274 

011895374592 
015055083468 

8.00 
9.00 

1.17109962*6 
1.4821729675 

-7834.2528 
8813.5344 

17.456 
19.638 

10  00 

1  82984317 

21.820 

022489692588 

11.00 

2.2141102353 

10772:0976 

24.002 

24  inches 

2.00  feet 

1.4142 

.0001414227124 

000565690849* 

1.00 
2.00 

013923055 

055692220 

1409.35008 

2818.70016 

3.1416 
6.2832 

!  0012728044116 
.0022623633984 
.0035355678100 
.0050912176464 

4^00 
5.00 
6.00 
7  00 

.125307495 

!  5012299  -i 

.t;>222'.»t')95 

4228.06024 

5637.40032 
7046.75040 
8456.10048 
9S65.  45046 

9.4248 
12.5664 
15.7080 
18.8496 
21.9912 

!  009061051899 

.0114552397044 

s'.oo 

9.00 

l'  127767455 

11274.80084 
12684.15072 

25.128S 

2-S.2744 

0141422712400 
.0171121482004 

10.00 
11.00 

1.39230550 
1.684689655 

1-1  CM.  500SO 
15502.85088 

31.4160 
34.5578 

SULLIVAN'S  NEW  HYDRAULICS 
TABLE  No.  50  CONCLUDED. 


5-31 


xl 


27  inches 
2.25  Feet 


30  inches 
2.50  Feet 


1.581 


.00047407 4C740 


.01)29(529629625 
.0042671671660 
.OJ5807 4074065 
.0375851851840 


0118518518500 
0143407407385 


.0001012018 
.0004048072 


0016192288 
0:)25  300450 
0036432648 


0064769152 
0081973458 
0101218000 
0122454178 

.0000769823 


0005928407 
.0012317168 
0019245575 
0027713628 
.0037721327 
0049268672 


1.00 
2.00 
8.00 
4.00 
5.00 
6.00 
7.00 
8.00 
9.00 
10.00 
11.00 

1.00 
2.00 
3.00 
4.00 
5.00 
6.00 
7.00 
8.00 
9.00 
10.00 
11.00 

1.00 
2.00 
3.00 
4.00 
5.00 
6.00 
7.00 
8.00 


.011721478 

.046885912 
.105493302 
.187543648 


.421973208 
.574352422 
.750174592 
.949439718 
1.17214780 


.1L941312 
.24908300 
.35867952 


.63765248 


1.20556172 


.00757891 
.03131564 


.18947275 
.27284076 


CO!  13148583 


.48505024 
.61389171 
.75789100 
.91704811 


1724.4288 
3448.8576 
5173.2864 
6897.7152 
8622.1440 
10346.5728 
12071.0016 
13795.4304 
15519.8592 
17244.2880 


2203.1592 
4406.3184 
6609.4776 


11015.7960 
13218.9552 
15422.1144 
17625.2736 

19828.4328 
22031.5920 
24234.7512 

3172.5672 
6345.1844 
9517.7016 


15862.8360 
19035.4032 
22207.9704 
25380.5376 
28553.1048 
31725.6720 
34898.2492 


3.976 
7.952 
11.928 
15.904 
19.880 

31 ! 808 
35.784 
39.760 
43.736 


14.727 
19.636 
24.545 
29.454 
34.363 
39.272 
44.181 


54.000 

7.069 
14.138 
21.201 


35.345 
42.414 
49.483 
56.552 
63.621 


77.759 


104.— Thickness  and  Weight  of  Cast  Iron  Pipe.— There 
is  a  great  want  of  uniformity  in  regard  to  the  thickness  of 
cast  iron  pipe  for  any  given  pressure.  Every  city  seems  to 
have  adopted  different  thicknesses  of  pipe.  The  leading  for- 
mulas for  thickness  give  greatly  differing  results  for  the  same 
conditions. 


SULLIVAN'S  NEW  HYDRAULICS. 

t=(.000058hd)  +.0152d+.312 (J.B.Francis 

t=(.0016  n  d)  +.013  d+.32 (M.  Dupuit 

t=(.00238  n  d)  +.34 (Julius  Weisbach 

The  following  formula  gives  thickness  of  cast  iron  pipe 
as  adopted  in  recent  practice, 

t=(p+100)  .000142  d+.33  (1— .01  d) (110) 

In  the  above  formulas, 
t=thickness  of  pipe  shell  in  inches 
d=inside  diameter  in  inches 
h=head  of  water  in  feet 
p=pressure  of  water=HX-434 
n=number  atmospheres  pressure  at  33  feet  each. 
Fannings  formula  for  the  weight  per  lineal  foot  of  cast 
iron  pipe,  including    the   weight  of    the  bell  or  hub  is,  for  12 
foot  pipes, 

W=12  (d+t)  Xl-08  tX3.1416X.2604 

By  a  12  foot  pipe  is  meant  a  pipe  which  will  actually  lay 
12  feet,  or  is  12  feet  from  bottom  of  bell  to  end  of  spigot.  The 
bell  or  hub  adds  about  1%  per  cent  to  the  weight  of  a  length 
of  pipe.  The  above  formula  allows  for  the  extra  weight  of 
bell.  For  more  on  weight  of  pipes,  see  "Gregory's  Practical 
Mathematics." 

105— Dimensions  and  Weight  ot  Cast  Iron  Pipe  Made 
by  The  Colorado  Fuel  and  Iron  Company  of  Denver,  for 
100  Ibs.  Pressure. 

TABLE  No.  51.          By  W.  F.    McCue 


I 

1 

»—  1 

1 
I 

Length  over  all. 
Feet  —  Inches. 

"3 

« 

•s 

a  I 
!l 

Will  Lay 
Feet—  Inches 

Thickness  of 
Shell—  Inches* 

Inside  Diameter 
of  Bell—  Inches 

s| 

is 

3 

0*0 

Outside  Diam. 
of  Spigot. 

«i 

fe 

M 

A  3 

9& 

*.s 

3 
4 

6 
8 
10 
12 
16 
20 

12-4 
12—4 
32—4 
12—4 
12—4 
12—4 
12—4 
12—4 

3 
3 
3 
3V4 
3K 
4 
4 

12—1 
12-1 
12-1 
12—  Vt 

1I-* 

12 
12 

1332 
7-16 
1-2 
17-32 
1932 
5-8 
34 
2732 

4  1-2 
5  1-2 

7  5-8 
9  7-8 
11  7-8 
133-1 
18 
22  1-8 

7 
81-4 
10  7-8 
13  38 
15  5-8 
17  3-4 
22  1-2 
26  7-8 

43-8 
5  3-8 
7  1-2 
9  3-4 
11  3-4 
13  5-8 
17  7-8 
22 

15  1-2 

33 
44 

63 

,1 

175 

*Sce  Table  No.  27.  §58  fo-  fractional  inches   in  equivalent 
decimals. 


270 


SULLIVAN'S  NEW  HYDRAULICS. 


Packing  (Jute  Hemp)  and  Lead  Required  Per   Joint  For 
Above  Pipe. 


Diameters  =* 

3" 

4' 

6" 

8" 

10" 

12" 

16' 

20" 

Lead,    Ibs.    Per 

Joint 

4  1-2 

5  1-2 

7 

10 

12  1-2 

17 

20  1-2 

29 

Packing,  ozs 

3 

3  1-2 

5 

7 

91-2 

12 

20 

26 

106—  Weight  Per  Foot  Length  of  Cast  Iron  Pipe  For  150 
and  200  Ibs.  Pressure,  as  Made  by  Colorado  Fuel  and  Iron 
Co.  of  Denver,  Colorado. 


Diameters 

3"    1    4' 

6" 

8- 

10" 

12" 

16" 

20" 

Wt.Pei.ft. 

150  pounds 
Pressure 

17  Ibs  23V,  Ibe 

361b< 

48Jbs 

70  Ibs 

85  Ibs 

140  Ibs 

210  Ibs 

Wt.  Per.  ft. 

200  pounds 
Pressure 

1 
19  lbs|26  Ibs 

421bi 

55  Ibs 

78  Ibs 

94  Ibs 

15b  Ibs 

222  Ibs 

REMARK— The  market  prices  of  pig  iron,  cast  iron  and 
lead  and  other  metals  fluctuate  so  rapidly  that  tables  for  esti- 
mating the  cost  of  pipe  and  laying  are  of  no  great  value  ex 
cept  in  so  far  as  such  tables  furnish  the  data  as  to  the  quan- 
tity and  weight  required.  The  price  of  pig  iron  May  12th, 
1898,  was  $6.65,  and  on  July  28th,  1899,  the  price  was  $15.25. 

The  present  price  of  cast  iron  pipe  (August  2nd,  1899)  is 
$33.00  per  ton  of  2,000  Ibs.,  and  of  lead,  $5.00  per  100  Ibs.  in 
Denver. 

107— Manufacturers'  Standard  Casllron  Water  Pipe 
For  100  Ibs.  Pressure  Per  Square  Inch. 

TABLE  No.  52. 


Diameters  In. 

4 

6 

8 

10 

12 

14 

10 

18      20 

24 

30 

36 

48 

Thickness,  In* 

1-2 

1-2 

1-2 

9-lfi 

9-16 

8-4 

8-4 

7  8  15-16 

1 

11-813-8 

11-2 

Wt.  per.  ft.  Ib. 

22 

88 

45 

ro 

75 

117 

125 

167    200 

251 

350 

475 

775 

Wt.per.   12ft. 

2ti4 

;«•*; 

540 

720 

900 

nm 

1500 

200012400 

no 

4200 

5700 

B800 

*See  Table  No.  27,  §  58  for  fractional  inches  in  equivalent 
decimals. 


108— Cost  Per  100  Feet  Length,  For  Labor  and  Ma- 
terial in  Laying  Cast  Iron  Water  Pipe  in  Denver,  Colorado, 
in  1890. 

The  conditions  were:— Top  of  pipe  5  feet  below  surface. 
Depth  of  trench  5  feet,  plus  outside  diameter  of  pipe.  Easy 


SULLIVAN'S  NEW  HYDRAULICS. 


271 


trenching  in  sandy  loam,  Wages,  foreman  $3.00,  calkere  12.50 
laborers  $1.75  per  day  of  10  hours,  teams  83.00  per  day,  pipe 
$33.00  per  ton  of  2000  Ibs..  lead  $4.15  per  100  Ibs.,  packing  6 
cents  per  Ib.  No  pavements  to  tear  up.  Backfilling  done  by 
teams  and  scrapers.  Average  water  pressure  80  Ibs.  Thick- 
ness of  pipe  for  120  Ibs.  hydraulic  pressure.  Hemp  packing. 
Hauling  pipe  60  cents  per  ton. 

TABLE  No.  53*  Cost  per  100  feet. 


I 


s, 


If 


22 
82 

4:, 
80 

7fi 
117 
125 

170 
'iJO 

a-,u 
r.oo 
no 


$86.30 

52.80 
74.25 
99.00 
123.75 
193.05 
206.25 
280. EO 
412.50 
577.50 
825.00 
1155.00 


10.20 


.94 
1.28 
1.88 
2.63 
4.50 
6.30 


$  0.66  $0. 

1.05 

1.35 

1.80 

2.25 

3.50 

3.75 

5.10 

7. tO 
10.  EO 
IB. 00 
21.00 


$  2.15 

3.13 

4.03 

4.14 

5.36 

8.07 

9.30 

11.17 

14.96 

16.15 

23.75 

49.87 


$0.20 
.20 

.20 
.'JO 
.20 


$0.15 
..15 
.15 
.20 
.20 
.25 
.25 

:! 

.60 

.70 

1.00 


15.00 
16.00 
20.00 
31.00 
32.00 
38.00 
44.00 
50.00 
60.00 
75.00 


*Allow  440  joints  per  mile  when  ebtimating  cost  of  lay- 
ing cast  iron  pipe.  Wrought  iron  and  steel  pipe  is  made  in 
lengths  of  15  to  27  feet  according  to  conditions  to  be  met. 
Cast  iron  pipe  is  in  lengths  of  12  feet.  See  Remark  under 
Table  No.  55. 

109— Cost  of  Pipe  Per  Foot  Laid  in  Boston. 

Axis  of  pipe  is  5  feet  below  surface.  Labor  $2.00  per 
day.  Cost  of  pipe  1^  cents  per  Ib.,  or  $30.00  per  ton  of  2000 
Ibs.  Special  castings  3  cents,  lead  5  cents  per  Ib.  Cost  of 
excavating  rock  $3.50  to  $5,50  per  cubic  yard,  measured  to 
neat  lines. 

This  table  is  transcribed  from  "Details  of  Water  Works 
etc.",  by  W,  R.  Billings. 


272  SULLIVAN'S  NEW  HYDRAULICS 

TABLE  No.  54.— Cast  Iron  Pipe. 


. 

d 

| 

1      11* 

5 

t—  1 

• 

| 

fc| 

"1  r-i 

"t£ 

_^ 

1 

| 

Jj] 

-J 

&. 

t3 

M    0      . 

&gj 

111 

bi 
a 

2  o 

J4^3 
0  jjfl 

§~ 

s 

.2 

i 

CUD" 
'S  u. 

££ 

T3  a 
a  P 
•go 
Hft 

I'3! 

O*0  CO 

M      *- 

UT3  I- 

£§£ 

1 

H.S 

•a- 

4 

6 

0.45 
0.50 

21.7 
85.0 

-0.70 
1.00 

$0.38 
0.57 

^S 

$0.02 
.08 

$0.25 
0.27 

$0.70 
.93 

8 

0.55 

50.0 

1.85 

0.83 

.08 

.05 

0.30 

1.26 

10 

0.60 

68.0 

1.70 

1.10 

.10 

.06 

0.34 

1.60 

12 
12 

0.58 
0.65 

S:S 

2.00 
2.00 

1.27 
1.42 

.13 
.13 

.07 

.07 

0.37 
0.37 

1.84 
1.99 

16 

0.66 

118.0 

2.70 

1.87 

.17 

.08 

0.45 

2.57 

16 

0.75 

185.0 

2.70 

2.12 

.17 

.08 

0.45 

2.82 

20 

0.73 

162.0 

3.35 

2.55 

.21 

.09 

0.55 

3.40 

20 

0.85 

183.0 

3.35 

2.94 

.21 

.09 

0.55 

8.79 

24 

0.81 

216.0 

4.00 

3.44 

.25 

.10 

0.68 

4.47 

24 

0.94 

250.0 

4.00 

3.95 

.25 

.10 

0.68 

4.98 

80 

0.93 

308.0 

5.00 

4.92 

.29 

.11 

0.80 

6.12 

36 

1.04 

410.0 

6.00 

6.58 

.34 

.12 

1.00 

8.04 

40 

1.12 

490.0 

6.70 

7.80 

.40 

.15 

1.30 

9.65 

48 

1.25 

660.0 

8.00 

10.40 

.48 

.20 

1.75 

12.83 

REMARK — From  the  high  cost  of  trenching  and  the  ref- 
erence to  rock  excavation,  the  ground  must  have  been  very 
"hard  digging."  Compare  cost  of  trenching  with  that  at 
Omaha  for  like  diameters. 

IIO—Cost  of  Trenching,  Laying,  Calking  and  Back- 
filling in  Omaha,  1889,  With  wages  of  Foreman  $2.50, 
Calkers  $2.25,  Laborers  $1.75  Per  Day  of  10  Hours.  (W. 

F.  McCue,  C.  E.,  of  Colorado  Fuel  &  Iron  Co.) 

TABLE  No.  55.— Cast  Iron  Pipe. 


Diam. 
of  Pipe 

Width 
of 
trench 
feet 

Depth  of 
trench, 
feet 

Cost  of 
trench, 
lineal 
foot 

Laying, 
calking, 
backfilling, 
lineal  foot. 

Cost  of 
labor  per 
lineal  foot 
complete. 

4' 

6 
8 
10 
12 
16 

1.75 
1.75 
1.75 
2.00 
2.00 
2.33 

5.666 
6.000 
6.000 
6.083 
6.250 
7.333 

$0.104 
0.10.1 
0.107 
0.126. 
0.126 
0.175 

$0.036 
0.036 
0.043 
0.053 
0.056 
0.063 

$0.140 
0.141 
0.150 
0.179 
0.182 
0.238 

REMARK — Mr.  McCue  lias  been  in    charge    of    the    con- 
struction of  nearly  700  miles  of  pipe  line  in   the  Eastern   and 


SULLIVAN'S  NEW  HYDRAULICS.  273 

Western  states,  In  a  letter  to  the  writer  he  says:  "We  gen- 
erally employed  60  to  70  men  in  a  gang— enough  laborers  to 
excavate  the  trench  ahead  of  the  layers.  In  laying  4  to  12 
inch  pipe,  we  had  one  yarner  aud  one  calker.  In  laying  pipe 
16  inches  diameter  or  larger,  we  had  two  yarners  and  two 
calkers.  In  laying  pipe  larger  than  12  inches  diameter  it  is 
necessary  to  use  a  derrick  for  lowering  the  pipe  into  the 
trench. 

One  yarner  and  one  calker  will  make  about  CO  joints  per 
day  of  10  hours  in  laying  4  or  6  inch  cast  iron  pipe,  and  about 
50  joints  of  8  inch,  45  joints  of  10  inch  and  40  joints  of  12 
inch  pipe.  In  laying  pipe  larger  than  12  inchs,  a  derrick  is 
required,  and  progress  is  much  less.  Most  of  the  backfilling 
is  done  by  team  and  scraper.  The  largest  days  work  I  ever 
had  done  was  80  joints  of  8  inch  pipe  yarned  by  one  man  and 
calked  by  one  man.  In  1893,  I  took  one  yarner  and  one 
calker,  the  fastest  I  ever  saw,  and  laid  and  calked  272  joints 
of  6  inch  pipe  in  35  hours.  The  cost  was  1^  cents  per  lineal 
foot  including  foreman,  kettlemen,  and  3  to  lay  pipe  in  the 
trench.  We  use  Jute  hemp  for  packing." 

III—Weston's  Tables  for  Estimating  Cost  of  Lay  lag 
Cast  Iron  Pipe. 

The  following  tables  by  E.  B.  Weston,  C.  E.,  of  Provi- 
dence, Rhode  Island,  were  published  in  Engineering  News, 
June  21, 1890,  together  with  other  valuable  data  of  like  char- 
acter. The  elements  of  cost  entering  these  tables  are: 

Wages,  foreman  $3.00,  calkers  $2.25,  laborers  $1.50,  per 
day.  Teams  $2.25  per  day.  Carting  $1.00  per  ton  of  2240 
Ibs.  Depth  of  trench  4.67  feet  plus  one-half  the  outside 
diameter  of  pipe.  Lead,  5  cents  per  pound.  Tools,  blocks 
and  wedges  7  2  10  to  16  1-10  per  cent  of  cost  of  trenching, 
laying  and  backfilling  the  trench.  In  the  tables  the  word 
"trenching"  includes  excavation  and  backfilling.  "Medium" 
digging  is  ground  equivalent  to  gravel  and  sand.  "Hard" 
digging  id  ground  equivalent  to  hard  or  moist  clay.  Cost  of 
engineering  and  inspection  not  included  in  tables. 


274 


SULLIVAN'S  NEW  HYDRAULICS. 


«s, 


g    §    8 


§     I 

"     § 


§§g 


1 1 


I  I 


3  |  § 


*H     ei 


§ 

S     ° 


111 


SSSS 


s  i  1  i  3  §  § 

-H    o    O    O    O    O    O 


S8SSS 


3  i 

3     e 


ifllilll 


SULLIVAN'S  NEW  HYDRAULICS. 


275 


112— Cubic    Yards  of  Excavation    in    Trench  Per   Lineal 
Boot,— Vertical  Sides— Bell  Holes  Mot  Included. 

TABLE  No.  57. 


Depth  of  Trench  in  Feet. 


10      11 


5.50 
6. 


8. JO 
9. 


77S 


0.370 
0.444 
0.518 


3.740 
0.815 
1.888 
1.968 

[.037 
1.111 

[.185 
[.259 
:.333 
1.407 


0.926 
015 
111 


0.888 
.000 

.111 


0.388  0.444 

0.518  0.592 

0.644  0.740 

0.777  0.888 
0.876 


3.000 
3.166 


i. :,:,:, 
0.740 
a.  926 
1.111 
296 
1.481 
1.666 
1.851 
2.037 
2.222 
2.407 
2.592 
2.777 
2.963 
J.148 
3.333 
3.518 


0.611 
0.815 
1.018 
1.222 
1.425 
628 
833 
2.037 
2.237 
2.444 
2.647 
2.851 
i.  or,:, 
3.258 
3.462 
3.666 
3.870 


0.666 
0.888 
1.111 

1.333 
1 . 555 

1.777 
2.000 

2' 222 


3.611 
3.851 
4.092 
4.000  4.333 
4.222  U.573 


0.722 

0.962 

203 

444 


2.648 


0.777 
037 

2.S8 

r,r,r> 

074 


2.852 

3.111 

.370 


4.148 

406 

4.666 

4.924 


REMARK.— The  foregoing  table  (No.  57)  will  be  useful  in 
estimating  the  cost  of  sewer  work  as  well  as  in  estimates  of 
cost  of  pipe  laying.  It  is  also  the  custom  of  some  engineers 
to  excavate  irrigation  canals  with  vertical  sides  and  allow  for 
the  caving  and  sliding  of  the  banks  until  they  assume  the 
natural  angle  of  repose.  There  is  nothing  to  commend  this 
practice,  but  still  it  is  followed  to  a  considerable  extent.  In 
sewer  work  where  the  ground  is  firm,  the  trench  is  excavated 
in  alternate  sections,  and  tunnels  driven  through  the  short 
blocks  of  ground  between  the  excavated  sections.  This  re- 
duces the  amount  of  excavation  by  about  30  per  cent,  and 
saves  the  cost  of  sheeting  and  bracing.  In  estimating  cost 
of  excavation  in  earth  or  rock,  see  Trautwine's  "Engineers 
Pocket  Book." 

113. -Bell  Holes  in   Trench  for   Cast  Iron   Pipes.— in 

order  that  the  "yarner"  and  the  calker  may  have  room  to  get 


276  SULLIVAN'S  NEW  HYDRAULICS. 

at  all  parts  of  the  joint,  the  trench  should  be  dug  out  8  inches 
deeper  for  a  distance  of  four  feet  in  front  of  the  bell  or  hub, 
and  8  inches  wider  on  either  side  for  the  same  distance  to 
give  shoulder  and  striking  room.  This  adds  materially  to  the 
cost  of  excavation,  especially  where  the  ground  has  a  tendency 
to  cave  and  slide,  or  is  very  wet. 

114  —Depth  of  Trenches  for  Pipe.— Pipes  in  which  there 
is  a  constant  flow  of  water  are  in  little  danger  from  freezing 
even  if  laid  on  the  surface  of  the  ground,  but  in  the  distribu- 
tion or  street  system  the  flow  id  almost  if  not  entirely  stopped 
during  certain  hours  of  the  night  when  little  or  no  water  is 
being  drawn  by  consumers. 

Pipes  supplying  reservoirs  and  having  a  constant  dis- 
charge may  be  covered  to  any  convenient  depth  simply  for 
the  protection  of  the  pipe  from  injury  by  wagons,  falling  trees 
etc.,  and  to  prevent  too  great  expansion  by  heat  or  con- 
traction by  cold,  and  to  get  the  pipe  out  of  the  way. 

The  general  rule  in  the  New  England  States  is  to  make 
the  trenches  for  street  pipes  of  such  depth  as  will  place  the 
center  or  axis  of  th3  pipe  five  feet  under  cover.  That  is,  the 
trench  is  five  feet  plus  one  half  the  outside  diameter  of  the 
pipe.  The  depth  that  a  street  pipe  should  be  covered  depends 
on  the  climate,  the  nature  of  the  ground  and  the  diameter  of 
the  pipe.  Where  the  temperature  gets  down  to  from  25  to  40 
degrees  below  zero  (Fahr.)  for  two  or  three  days  at  a  time,  4 
and  6  inch  pipes  will  freeze  solid  when  five  feet  under  cover 
in  sandy  and  gravelly  loam.  This  occurred  in  many  towns  in 
Colorado  in  February,  1899.  If  the  earth  id  dense  and  free 
from  stones  and  gravel  it  is  not  probable  that  frost  will  pene- 
trate to  a  depth  exceeding  four  and  a  half  feet.  Small  pipes 
laid  in  open,  gravelly  soil,  should  have  the  top  of  the  pipe  at 
least  six  feet  under  cover. 

115.— Amount  of  Trenching  and  Pipe  Laying  Per  Day 
Per  Man. — The  number  of  cubic  yards  of  excavation  done 
per  man  per  day  will  be  less  in  deep  trenchee  than  in  com- 


SULLIVAN'S  NEW  HYDRAULICS.  277 

paratively  shallow  ones  because  of  the  extra  effort  required 
to  throw  the  dirt  out  of  deep  trenches.  The  nature  of  the 
earth  or  rock  to  be  excavated  will,  of  course,  be  a  controlling 
element  in  determining  the  amount  of  excavation  that  can  be 
accomplished  per  day,  by  an  average  laborer.  Quicksand, 
water  and  caving  banks  may  also  be  large  items  of  expense 
and  prevent  rapid  progress.  There  are  so  many  elements  of 
uncertainty  invol/ed  in  making  an  estimate  of  the  work  that 
one  man  will  accomplish  in  a  given  time  that  it  is  best  to  as- 
certain what  has  been  actually  accomplished  under  like  con- 
ditions in  the  past.  By  analysis  of  statements  of  work 
actually  done  in  a  given  time  by  a  given  number  of  men,  we 
can  approximate  the  time  required  and  the  cost  of  doing 
similar  work.  Mr.  W.  R.  Billings,  superintendent  of  the 
Taunton,  Mass.,  Water  Works  (1887)  says*:— 

"The  following  notes  of  actual  work  are  offered,  not  in 
any  sense  as  instances  of  model  performance,  but  as  simple 
illustrations:  Time  July  6th  1887;  gang  60  men,  16  inch  pipe, 
2  yarners,  2  calkers,  4  to  10  men  digging  bell  holes,  30  boll 
holes  per  day,  400  feet  of  pipe  laid  and  jointed  in  10  hours." 

These  notes  are  somewhat  incomplete  in  that  they  do  not 
disclose  the  following  items: — (1)  nature  of  earth  excavated; 
(2)  depth  of  trench;  (3)  width  of  trench;  (4)  what  part  of  the 
total  400  feet  length  of  trench  and  bell  holes  made  on  July 
6th.  (5)  Was  the  trench  back-filled  for  400  feet  on  July  6th. 
(6)  How  many  of  the  60  men  were  in  the  derrick  gang.  (7) 
Did  the  derrick  gang  assist  in  excavating  a  part  of  the  400 
feet  of  trench  before  beginning  to  lay  pipe,  or  was  a  part  of 
the  trench  and  bell  holes  made  on  the  day  before.  Mr.  Bil- 
lings statement  shows  that  4  to  10  men  working  10  hours 
made  30  bell  holes  for  16  inch  pipe.  30  bell  holes  would  ac- 
commodate only  360  feet  of  12  foot  pipes.  He  states  that  400 
feet  were  laid.  It  is  therefore  evident  that  some  part  (at 
least  40  feet)  of  trench  and  bell  holes  must  have  been  made  on 
some  other  day. 

*Details  of  Water  Works  Construction,  p.  55.  (Published  by  "Engi- 
neering Record"  N.  Y.) 


278  SULLIVAN'S  NEW  HYDRAULICS 

In  another  chapter  of  Mr.  Billings  work  we  find  some 
"Notes  on  the  construction  of  two  miles  of  16  inch  water 
main,"  in  1887.  The  date  shows  that  it  is  the  same  pipe  above 
referred  to.  From  these  notes  we  gather  the  following  facts: 
The  pipe  was  hauled  an  average  distance  of  1)^  miles  over 
good  roads  for  64  cents  per  ton  of  2240  Ibs.  The  first  division 
of  the  pipe  line  was  2,927  feet  in  length.  The  trenching  was 
in  good  ground  except  a  short  stretch  of  quicksand  and  water 
The  total  cost  of  labor  for  this  division  of  the  line  was  32.30 
cents  per  lineal  foot,  including  all  labor  charged  on  the  time 
book  from  foreman  to  water  boy,  in  a  gang  of  60  men.  An- 
other division  of  2,100  feet  length  furnished  sandy  digging 
with  some  tendency  to  caving.  A  brook  had  to  be  crossed 
and  a  blow-off  located  which  required  the  trench  to  be  10  or 
12  feet  deep  for  100  feet  length.  An  old  8  inch  pipe  had  to 
be  removed,  and  18  services  furnished  with  a  temporary  sup- 
ply. The  cost  of  labor  per  lineal  foot  for  this  division  was 

34.7  cents.    In  the   next  division   the  digging  was   dry  and 
sandy,  and  caving  of  the  trench  was  almost  constant.    An  old 
8  inch  pipe  had  to  be  taken  up,  and  a  temporary  supply  main- 
tained for  53  services.    The  cost  of  labor  on  this  division  was 

41.8  cents  per  lineal  foot. 

In  the  next  division  the  digging  was  wet  and  dirty.  Old 
pipe  had  to  be  taken  up,  and  temporary  supply  maintained 
for  30  services,  and  four  connections  made  for  a  manufactur- 
ing company.  The  cost  of  labor  in  this  division  was  47.4 
cents  per  lineal  foot.  The  mill  connections  being  the  princi- 
pal cause  of  the  increased  expense. 

Mr.  Billings  states  that  a  detachment  of  the  same  gang 
of  men  laid  2,000  feet  of  8  inch  pipe  in  new  ground,  good  dig- 
ging, at  a  cost  of  17.3  cents  per  lineal  foot  for  all  labor,  and 
1060  feet  of  4  inch  pipe  at  a  cost  of  13.10  cents  per  foot,  and 
600  fee't  of  6  inch  pipe  at  15.38  cents  per  lineal  foot. 

While  the  depth  and  width  of  trench  and  daily  wages 
paid  are  not  stated,  it  will  be  near  enough  to  assume  that  the 
trenches  were  5  feet  plus  one  half  the  diameter  (outside)  of 
the  pipe  to  be  laid,  and  the  width  of  trench  2.333  to  3.00  feet, 


SULLIVAN'S  NEW  HYDRAULICS.  279 

according  to  size  of  pipe  (4*,  6",  8"  and  16"  diameters).  For 
amount  of  excavation  in  bell  holes,  refer  to  paragraph 
110,  ante. 

Assume  wages  as  follows:  Foreman  $3,  Calkers  and 
yarners  $2.25,  Derrick  gang  (6  to  10  men)  $1.75,  laborers  $1.50 
per  day  of  10  hours.  A  gang  of  six  men  is  sufficient  to 
handle  the  4,  6,  and  8  inch  pipe,  together  with  one  yarner  and 
one  calker.  For  the  16  inch  pipe  it  will  require  2  yarners 
and  2  calkers  and  10  men  in  the  derrick  gang.  The  remain- 
der of  the  gang  of  60  men  will  be  laborers  digging  trench 
and  bell-holes  ahead  of  the  derrick  gang.  Subtracting  the 
number  of  calkers  and  yarnere  and  derrick  gang  and  the 
foreman  from  60,  the  remainder  shows  the  number  of  men  en- 
gaged in  trenching  and  digging  bell-holes.  The  length  of 
trench  and  bell-holes  completed  in  10  hours  gives  a  basis  of 
calculating  the  cubic  yards  excavated  by  each  man  per  day, 
and  the  wages  paid  him  per  day  furnishes  the  data  for  find- 
ing the  cost  per  cubic  yard  of  excavation.  In  fairly  good 
digging  it  will  be  found  that  one  man  will  make  from  5.60  to 
6.25  cubic  yards  of  excavation  per  day  of  10  hours.  In  ex- 
cavating rock  the  average  will  be  from,  .50  to  1.50  cubic  yards 
of  excavation  per  man  per  day,  depending  on  the  nature  of 
the  rock.  The  excavation  of  deep,  narrow  trenches  is  very 
much  more  expensive  per  cubic  yard  than  railroad  and  canal 
work  in  like  earth  or  rock.  See  "Remark"  under  table  No.  55, 
Section  107.  With  labor  at  $2  per  day  the  cost  of  excavation 
in  rock  was  $3.50  to  $5.50  per  cubic  yard,  measured  in  place, 
in  the  City  of  Boston.  This  was  an  average  of  from  .3636  to 
.57  cubic  yard  of  excavation  in  rock  per  man  per  day.  In 
very  wet  trenches  the  digging  of  sumps,  sheet  piling  and 
bracing,  and  pumping  out  of  water  is  a  heavy  expense  in  ad- 
dition to  the  ordinary  cost,  and  will  amount  to  from  40  cents 
to  $1.00  per  lineal  foot. 

By  reference  to  table  No.  56,  it  will  be  seen  that  the  cost 
of  laying  4",  6%  8"  and  16"  pipe  as  given  above  by  Mr.  Billings, 
is  about  the  same  as  given  in  Weston'a  Table  for  "medium" 
digging,  and  also  about  the  same  as  shown  in  table  No.  53  for 


280  SULLIVAN'S  NEW  HYDRAULICS. 

cost  of  laying  pipe  in  Denver,  in  sandy  loam.  Referring  to 
cost  of  trenching  in  Omaha  (table  No.  55)  with  wages  of 
laborers  at  $1.75  per  day,  and  we  find  that  a  trench,  in  good 
digging,  5.666  feet  deep  and  1.75  feet  wide,  cost  .10^  cents  per 
lineal  foot.  In  a  lineal  foot  of  this  trench  there  were  5.666X 

1.75-i-27=.36724  cubic  yards  of  excavation,  or  l  =2.723  lin- 
eal feet  of  trench  to  the  cubic  yard  of  excavation.  With  the 
cost  at  .104  cents  per  lineal  foot  of  trench,  and  2,723  lineal 
feet  to  the  cubic  yard,  the  cost  per  cubic  yard  of  excavation 
was  2.723  X.104=.284  cents. 

With  wages  at  $1.75  per  day  of  10  hours,  the  average  work 
done  by  one  man  in  one  day  was  '  =6.162  cubic  yards. 

One  man  would  therefore  average  2.723x6.162=16.779  feet 
length  of  trench  of  those  dimensions  and  in  that  kind  of 
ground,  per  day.  If  wages  were  reduced  to  $1.50  per  day,  the 
cost  of  trench  would  be  \  part  less,  or  .24343  cents  per  cubic 
yard  of  excavation.  In  stiff  clay  or  cemented  gravel,  one  man 
will  average  about  4.50  cubic  yards  of  excavation  per  day,  and 
the  cost  at  $1.50  per  day  wages,  will  be  33%  cents  per  cubic 
yard,  or  if  wages  are  $1.75  per  day,  the  cost  will  be  .39  cents 
per  cubic  yard  of  excavation.  Hence  in  stiff  clay  or  cemented 
gravel  the  average  progress  per  man  per  day  would  be 
12.25  lineal  feet  of  trench  5.666  feet  deep  by  1.75  feet  wide. 
Trenches  for  larger  diameters  than  8  inches  would  be  both 
deeper  and  wider,  and  the  cubic  yards  of  excavation  per 
lineal  foot  would  be  increased  in  proportion. 

One  yarner  and  one  calker  will  joint  cast  iron  pipe  about 
as  follows,  in  an  average  days  work:  720 feet  of  4  inch  pipe, 
or  660  feet  of  6  inch,  or  600  feet  of  8  inch,  or  540  feet  of  10  inch, 
or  480  feet  of  12  inch,  or  360  feet  of  14  inch,  or  200  feet  of  16 
inch  pipe. 


SULLIVAN'S  NEW  HYDRAULICS.  281 

The  number  of  joints  made  will  depend  on  whether  the 
trench  is  wet  or  dry  or  stands  up  well  or  caves.  The  wages  of 
calkers  and  yarners  are  usually  from  50  to  75  cents  per  day 
more  than  the  wages  paid  to  ordinary  laborers. 

In  estimating  the  cost  of  completing  a  cast  iron  pipe  sys- 
tem add  five  per  cent  to  cost  of  the  pipe  in  order  to  allow  for 
breakage.  Also  add  the  cost  of  engineering  and  inspection. 

116— Lead  Required  Per  Joint  For  Cast  Iron  Pipe. 

The  quantity  of  lead  required  per  joint  for  cast  iron  pipe 
depends  on  the  dimensions  of  the  lead  space  between  the 
bell  and  spigot,  and  also  upon  the  manner  in  which  the  joint 
is  yarned  or  packed.  There  is  no  uniform  rule  observed  in 
the  manufacture  of  cast  iron  pipe  as  to  the  dimensions  of  the 
lead  space,  and  consequently  no  rule  can  be  framed  for  de- 
termining the  quantity  of  lead  required  per  joint.  The  in- 
ner diameter  of  some  bells  is  uniform  while  in  others  it  con- 
verges. Inside  of  some  bells  there  is  a  groove,  semi-circular 
in  form,  extending  around  the  inner  circumference  of  the 
bell.  Others  are  plain  without  grooves.  Different  foundries 
adopt  different  depths  and  slopes  of  the  lead  space,  and  some 
yarners  will  put  twice  as  much  yarn  into  a  joint  as  others. 
Some  engineers  adopt  the  rule  of  estimating  2  Ibs.  of  lead 
for  each  inch  diameter,  as  being  approximately  the  quantity 
required  per  joint.  The  result  of  this  rule  is  too  much  for 
small  diameters  and  not  enough  for  large  diameters.  The 
amount  of  lead  per  joint  used  in  recent  practice  is  from  one. 
third  to  one-half  less  than  formerly,  and  the  tendency  is  to 
reduce  the  quantity  still  more. 

In  laying  6",  8"  and  16"  pipe  in  Taunton,  Mass.,  in  1887, 
Mr.  Billings  used  7.68  Ibs.  per  joint  for  6  inch  pipe,  9.12  Iba., 
per  joint  for  8  inch  and  21  Ibs.  per  joint  for  16  inch  pipe. 


282 


SULLIVAN'S  NEW  HYDRAULICS 


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If  belle  and  lead  spaces  were  designed  and  specified  by 
engineers  with  regard  to  flexibility  of  joint  and  economy  in 
the  quantity  of  lead  and  packing  required,  one  third  the  lead 
now  used  would  be  saved  and  the  joints  would  be  equally 
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SULLIVAN'S  NEW  HYDRAULICS.  283 

orally  agreed  that  the  calking  tool  does  not  affect  or  set  up 
the  lead  to  a  greater  depth  than  one  inch,  and  the  lead  be- 
yond this  is  worse  than  wasted,  for  it  only  stiffens  the  joint, 
which  is  really  an  element  of  weakness. 

AB  the  circumference  of  a  pipe  increases  directly  as  the 
diameter,  and  as  there  is  no  sufficient  reason  for  increasing 
the  depth  of  bell  as  the  diameter  increases,  there  is  no  reason 
why  the  quantity  of  lead  required  per  joint  should  not  vary 
directly  with  the  diameter  if  the  bell  and  spigot  were  prop- 
erly designed. 

Three  inches  is  an  ample  depth  of  bell  for  any  diameter, 
and  greater  than  necessary.  Unequal  settlement  in  the 
trench  must  be  prevented  by  proper  laying  anyway. 


SULLIVAN'S  NEW  HYDRAULICS. 


INDEX. 


A 

PAGE. 

Abbot,  Gen.  H.  L.,  discussion  of  data 84,  Bee  also  23, 30,  54,  98,  97 

Abbot,    Humphreys   and,  River  Formula, , 11 

Abrasion  of  Banks  by  frost  and  flow gl,  121,189.  222,  229 

Acceleration   of  gravity 21,22,31,  32,60,  61,62,66,99,215,216 

Acre,  area  of,  in  square  feet 166 

Action  and    reaction  at  bends 194,217,223 

Action  of  frost  on  canal  banks 81,  121,  222,  223,  228,  229 

Adjustment  of   side    slopes   of  canals 228,231 

Air,  friction  of,  at  water  surface 216,  217 

Alignment  of  canal  or   flume 69,  96,  97, 124, 125,  223,  224,  229,  234,  235 

Altitude  affect 3  g  in  the  formula  99 

Amount  of  pipe  laid  per  day 276,277,  281 

Angle  of  bend  in  canals 186, 187, 189,  213, 217,  223,  224,  229 


Angleof  bend  in  pipes 186,  187,188,189 

Angle  of  repose  of  earth 229,  230 

Appendix  I.     New   Orifice  and    Weir  formulas 242,  249 


Appendix  II .    Water  works  general  data  and  f 01  mulas 249, ' 

Application   and  limitation  of  formulas 6,36,37,101,105,232 

Aqueducts 27,  28,29,09,70,96,  97,124,125 

Area  of  canal,  how  found 119, 120 

Area  of  canals,  tables  of 121,125 

Area  of  egg-shaped  sewers  112, 113 

Area  of  nozzle  discharge;  how  determined 171, 173, 182 

Area,  of  one  acre  in  square  feet 166 

Area  of  rivets  to  net  pipe  metal 200  201,  202 

Area  of  channel  varies  with  square  of  friction  surface,  34,  36,  39,  40,  49 

104,105,  238,  239 

Ashlar  masonry,  coefficient 69,  70,  74.    See  28,  29,  58,  240,  241 

Asphaltum,  density  of 57 

Asphaltum  coated  cast  iron  pipes,  coefficient 63,  61,  67, 151, 15:5,  260,  262 

Asphaltum  coated  riveted  pipe,  coefficient,  63,  64,  67, 110,  111,  151, 152,  153, 

186,  197.  260,  262 

Asphaltnm  coated  welded  pipe,  coefilcient 64, 185, 197,  260,  262 

Asphaltum  coating,  how  made  and  applied Ill,  179, 185 

Average  consumption  of  water  in  towns  and  cities 249,  252 

Average  of  roughness  of   perimeter 28,58,  80,81,  240,241 

Average  value  of  C  for  irregular  perimeter 28,  48,  58,  75,  76,  81,  210,  241 


Banks  of  channel,  density  of,  affects  coefficient 1..54,  96,  97 

convergent  and  irregular,  6,  28,  29,  58,  76,  81,  84,  212,  213 

......I........ .....214,  224,  236 

Bazin's  formula 10 

Bell  holes  in  pipe,  trench 275,276 

Bends  in  open   channels 186,  188,  189,  213,  217,  223,  224,  229 

Bends  in  pipes 186,  187,188 


INDEX.  285 

PAGE. 

BlackwelTs  formula 9 

Bodies,  law  of  falling 31,32.60,61.62,66 

Boston,  cost  of  pipe  laying  in 271,  272 

Bottom  velocity,  controlled  by  grade 220,  221,  223,  225 

grade  required  for 226,228,230 

formula  for 225,226,228,230 

which  erodes  and  moves  material 227 

value  of  R  does  not  affect 219,  220,223,  225 

ratio  to  other  velocities 218, 219.  220,  22(5,  236 

Brahms  &  Eytelwein's  formula 10 

Brass  fire  nozzles 140  143 

Brass  pipes,  coefficient .".'.' .'.'.'.'l4i,'l42, 144 

Brick  conduits 27,28,29,70,  71,72,73,111,112 

Brick /ewers HI,  112,113,114 

Bursting  pressure 52,195,196 


Calculating  weight  of  riveted  pipes 206,  207 

Canal,  area  of  cross-section,  how  found 119, 120 

areas  of.  table 121,125 

alignment 69,  96.  97, 124,  125,  223,  224,  229,  234,  235 

banks,  affected  by  frost 81, 121,  222,  223.  228,  229 

banks,  convergent  and  irregular  .  ..6, 12,  27,  28,  58,  213,  221,  240,  241 

banks,  density  of,  affects  coefficient 54,  55,  56,  57,  67,  96,  97, 124 

benda  in 69,  96,  97, 124, 186, 188, 189,  213,  217,  223,  22 1,  229 

bed,  stability  of 222,226,227,230 

bottom  velocity  in,  controlled  by  grade 220,  221,  223,  225 

bottom  velocity,  eroding  power 227 

bottom  velocity,  formula  for 225,  226,  228,  230 

bottom  velocity,  grade  required  for  given 226,  228,  230 

bottom  velocity,  no  ratio  to  others 218,219,220,  226,236 

bottom  velocity,  size  of  canal  does  not  affect 219,  220,  223,  225 

Coefficients,  tables  of ....87,96 

coefficients  vary  with  mean  roughness,  27,  28,  29,  80,  58,  75,  76.  81 240 

dimensions  of,  how  found . Ill "  II"  "II 1 1 II 1 1 II II I  .'I . iJ9,  i20J  121, 125 


djmensions  to  carry  given  flow 231,232 

dimensioi  s,  formulas  for  R,  p  and  a.       .  231,232 

effective  value  of  grade  or  sk  pe 237,  238,  239,  241 


efficiency  of 

erod  ng  velocity  in 222,223,224,229 

evaporation  and  leakage 232,233 

excavation,  table 275 

See  also 121,125 

float  measurement  of  velocity  in 68,  96,  97,  235,  236 

See  also ....24,25 

flow  in 212,236 

See  also r.8,  210,  241, 121, 125 

flume  fo  mirgpart  of 234 

See  also 07,  68,  69,  96,  97, 124,125 

formula  for  flow 105,241 

Seeabo 124,125 

grade  for   bottom  velocity 226,228,230 

grade  for  mean  velocity 120, 121,128 

grade,  value  of,  varies  with  y'l  3 237,  239,  211 

hj  dran  ic  mean  depth,  how  1'ound 119,  120, 121,  231,  232 

length  in  which  is  on.,  loot  fall 121,  125, 128,  181 

mean  velocity  formulas 52,53,98,105,124,  J2r>,  241 

mean  velocity  in 221,  222,  235,23.1 


286  SULLIVAN,S  NEW  HYDRAULICS. 

PAGE. 

Canal,  new,  flow  in 92.232,233 

new  losses  from 232,233 

perimeter,  how  found 120,  231,  232 

perimeter,  roughness  defined 96,  97 

Seeal-o 27,28,29,58,240.241 

quantity  discharged  by 121, 125,  235,  236 

ratio  of  velocities  in 218,  219,  236 

See  also 25 

reaction  ae  bends  ia 189,217,  223 

reactions  at  perimeter 218,  221 

return  seepage 233 

Bide  slopes 222,  223,228,229,230 

slope  of,  for  bottom  velocity 226,  228,  2EO 

slope  of,  for  mean  velocity 120,  lv;l,  128 

slope  or  grade  varies  in  value  with  -j/r3 237.239,241 

slope,  scouring  velocity  depends  on 220,  227 

surface  velocity 216,217,  221,  222,  236 

stability  of  bed 222,226,227,230 

whirls  and  boi's  in 217 

wind,  effect  on  flow 217 

Capacity,  relative,  of  pipes 101 

Cast  iron,  specific  gravity  of 57 

Cast  iron  nozzles  and  giants 140, 171,  180, 181, 182, 183,  210,  211 

Cast  iron  pipes,  densit  y  of  affects  coefficient 54,  57 

cost  of  laying 270.281 

dimensions  and  weight 269,  270,  271 

dimensions  and  contents 162, 163 

joints 209,270,281.282,283 

tensile  strength 198 

thickness  of  eht-li 197,198,268,269 

weight  of 269,  270,  271 

Cement  joints,  vitrified  pipes 161, 162 

Cement  linings,  coefficient 55,  66,  69,  71 

Cement  mortar  linings,  coefficient 55,  66,  67 

Census  of  cities,  water  supply 249,  252 

Circle,  properties  of, 34,  35 

( ' i ties,  water  supplied  to  various 219, 252 

Cities,  cost  of  water  in 251,252 

Clay  roll  for  running  lead  joints 209 

Coating  for  water  pipes 111,179,180,185 

Coefficients,  of  entry 7,59.60,62 

flow,  or  velocity 38,  51,  54,  67,  99, 101, 102,  259,  260 

friction  or  resistance. . .  .37,  48,  54, 98,  )01, 102,  261,  262 
Coefficients,  are  constant,  29,  30,  37.  38,  39, 42,  47,  48.  58,  62,  65.  66,  138,  240, 

241 

are  correct  index  of  roughness 42.  47,  48,  240,  241 

conversion  of 51,53,101,  128,  155,162 

density  of  perimeter  affects 54,96,124,125 

discussion  of 237,241 

Seeaho 34  to  54 

in  Chezy  or  Kutter  formula 6,  7,  42,  43,  44,  81,  241 

in  terms  of  cubic  feet 51,  100 

in  terms  of  diameter  in  feet  and  head 98,106 

in  terms  of  diameter  in  inches 51, 118 

in  terms  of  diameter  and  pressure 101 , 102 

in  terms  of  gallons  per  minute 51, 137 

in  terms  of  hydraulic  m<  an  depth 103, 104,  106, 107 

orifice 7,  96,97,246.247,248,  249 

reverse  variation  of 27:  28,  29,  30,  58,  240,  241 


INDEX.  287 

PAGE. 

Coefficients,  varies  only  with  roughness 240,  241 

See  also 58,23,29 

for  aqueducts.... 27.  28,  29,  58,  67,68.  69,  70,  71,  124, 125,240,  241 

asphaltum  coated  cast  iron  pipes 260,  262 

See  also 63,64 

asphaltum  coated  riveted  pipe 57, 67, 144, 151,  260,  262 

brass  fire  nozzles 142, 143. 144,  210,  211 

brass  pipes 141,142.144 

brick  conduits,  27,  28,  29,  58,  69,  70,  71,  72,  73,  111,   112,    113, 

124,  125 

cast  iron  pipes 52,  54,  55,  64,  C5,  67, 108, 150,  260,  262 

cast  iron  nozzles 6,146,210.211 

cement  lined  pipe 55,66.69,71 

cement  mortar  lined  pipe 55,  66,  67,  260,  261 

convergent  pipes 142, 143, 146,  210, 211 

converse  lock-joint  pipe 185, 186,  209 

concrete  conduits 73 

conduits  of  wood 07,  68,  69,  lt>0, 161, 124.  125 

cansls 87,96 

See 27,28,29,58,240,241 

different  depths  of  How 28,  29,  58,  240,  241 

earthenware  pipes 161, 162 

flumes 67,  68,  C9,  fc6, 124, 125,  234 

hammer  dressed  btor.e 70,  75 

lead  pipes 57,59.67,193 

rivers 58,  fcl,  96,  240,  241 

rough  masomy 74,  75,  76,  86 

rubble  masonry 76.77,  78,  92 

rubber  fire  hobo 137,138,140 

rubber  pipes 144 

vitrifiu  d   pipes 161,162 

wooden  pipes 67,68,69,160,161,124,125 

wiers  and  oiific's 242,249 

Combination  of  different  roughnesses  «.f  verimeter,  27,  28,  29,30,  58,  240, 

Common  properties  of  pipes  and  open  channels 36.  37, 104, 105,  231,  232 

Concrete  conduits 73 

Conical  pipes,  friction  in 140,    146, 172, 181, 182, 183, 210,  211 

Consumption  of  water  in  citits 251,252 

Contents  of  pipe  per  foot 162, 163 

Contracted  vein,  form  of 143,210,211 

Convergent  pipes,  nozzles  and  reducers 143, 146,  210,  211 

See 163,171,180 

Conversion  of  coefficient  s 51,  53, 101, 128, 155, 162 

Conversion  of  U.  S.  to  Metrical  measures 165,166 

Correction  of  text 237,238,239,240,241 

Cost  of  pipe  laying  in  Boston 271,  272 

Denver 270,271 

Omaha 272,273 

Weston's  tables  for 273,274 

Cost  of  trenching i 270,  276,  281 

Cost  of  water  per  10CO  gallons 251,252 

Cox's  formula ' 

Cubic  foot  of  water 167 

Cubic  measure 16  < 

Cubic  feet  in  equivalent  gallons 256,  259 

Cubic  feet  per  second,  logarithms  of 2at>,  £>9 

Cubic  yards  of  excavation  per  man 276,  277,  281 

Cubic  yards  of  excavation,  table  of ••    273 

Current  meters....  24,71,96,97 


SULLIVAN'S  NEW  HYDRAULICS 


PAGE. 

D'Arcy's  formula 8,11 

D' Arcy  &  Bazin's  formula 11,  22,  23 

D'Arcy— Pitot  tube  gaugings 68,74 

Data  of  flow  defective 23  to  30,  58, 60,  61,  62,66,  72,81,  84,  96,  97 

Data  of  flow  forming  basis  of  author's  formula 65, 137. 138 

Data  of  flow,  how  to  test 60, 61,  (Compare  with  65, 138) 

Data  of  flow  in  pipes,  conduits  and  rivers 58  to  96 

Data  of  water  supply  in  cities 249,252 

Data  of  water  works  construction 249,283 

D'Aubuieson's  formula 9, 11 

Days  work  of  one  man '. 276,  277,281 

Decimal  parts  of  inch  and  f  iot 164,205 

Defects  in  old  formulas  for  fl  . w,  3,  8,  23,  37, 3<,  43,  44,  48,  61,  62,  66,  237,  241 

Defects  in  sheot  metal,  how  detected 208 

Defects  in  weir  aud  orifice  formulas 6,  7,  64,  68,  96,  97,  242  to  249 

Density  of  perimeter  affects  coefficient 54,  57,  95,97,124,125 

Denver,  Colorado,  cost  of  pipe  laying  in 270,  271 

Depth,  hydraulic  mean,  of  canal,  how  found i20,  231,232 

egg-shaped  sewers 112, 113 

Depth  of  penetration  of  frost 276 

Depth  of  pipe  trench 276 

Diameter  found  from  area 171, 172, 173 

Diameter  of  giant,  nozzle,  or  orifice,   how  found, 171,172,173,210  211 

(Sao  140  to  148) 

Diameter  of  cast  iron  pipe  for  given  discharge,  100,  108,  109, 136, 149.  161, 

184,254,259260,263 

Diameter  of  coated  pipe  forgiven  discharge 100,  102, 110,260 

Diameter  of  cement  mortar  lined  pipe  for  given  discharge 260 

Diameter  of  earthenware  pipe  for  given  discharge 161, 162 

Diameter  of  steel  pipes  forgiven  discharge 260 

Diameter  of  welded  pipes  for  given  discharge 260 

Diameter  of  wooden  pipes  forgiven  discharge 260 

See 160, 161,  and  67.68 

Diameter  of  pressure  or  power  mains 170,  171, 181. 185, 186,  261.  262 

(Sen  100, 149, 150) 

Dimensions  of  canals 120, 124, 125,  23',  232 

cast  iron  pipes 162, 163,  269,  270,  271 

cast  iron  gi-mts 146,  210,211 

fire  nozzles 140. 143.  210,  211 

flames 124.125 

reducers 210,211 

sheet  metal  for  given  diimeter 2  6 

sewers : Ill,  112,113 

Discharge  area  of  nozzle 171, 173, 182 

brick  sewers 112,113 

canals 121,124 

cast  iron  pipes 108, 109, 127, 129.  J50,  151,  268,  264,  268 

coated  riveted  pipes 110,  111,  151, 152, 153,  2BO 

cement  lined  pipes 260 

See T.:. 55,66,67,69,71 

earthenware  pipes Ml, 162 

flumes 124,  125,234 

wooden  pipes 67,  68,  69, 160, 162,  124, 125,  260 

E 

Earth,  angle  of  repose 229,230 


INDEX.  289 

PAGE. 

canals  in 87,96,121,124,212 

,_    (See  12,  54, 57,  58,  241) 

Earthenware  pipes 161,162 

Effective  value  of  the  slop- 48,237,241 

Efficiency  of  canal 228 

Efficiency  of  water  wheel 169,183 

Egg-shaped  sewers Ill,  112.  113 

Elementary  dimensions  of  pipes 162, 183 

Eleventh  roots  and  powers 184 

Entry,  resistance  to 7.  59.  60,  62 

Equilibrium  of  gravity  and  resistance 22,24,27,28,  37,  b2,  215,  216 

Eroding  velocity 222,223,227,229 

Erroneous  data  of  flow,  example  of 60,  6l,  62 

See  23  to  30 

Errors  in  formulas  for  flow 3,  8, 13,  23,  37, 42,  44,  61,  62,  6<j,  2*7,  241 

Errors  in  orifice  and  weir  formulas 6,  7,  64,  68,  96,  97,  242,  249 

Evaporation  and  leakaee  from  canals 232,233 

Excavating  pipe  trenches,  cost 270  to  281 

Excavation,  table  of 275 

Experiments  forming  basis  of  our  formula 65,138 

F 

Factor  of  safety 197 

Fall,  distance  for  one  foot 121, 125, 128, 181 

Fall,  head,  or  slope 99,128,157,158,161,169,180 

Fulling  bodies,  law  of 31,32 

Kire  hose 137,146 

Fire  hydrant  presi-ure 139, 154,155,  164 

Fire  nozzles 140,    146,210,211 

Floats,  d-.uble 25,96,17,235.236 

surface 25,96,97 

Flow,  discus  ion  of '. 34,  to  58,212,  237,  241 

S*e21to30 

generwl  formulas  for 98,  JOS,  241,  259,  2  0,  261.  262 

in  silted  conduits 27,  28,  29,  30,  58,  237,  241 

in  canals  and  river- 21,  30. 48.  54,  57,  58,  81,  212,  237,  241 

permanent  and  uniform 27,  37,  212 

of  a  par  icle  of  water 218 

resi=t  nee  to.  directly  as  roughness,  22,  23,  32,  33,  48,  58,  237,  239,  241 

resistance  to,  inversely  as^/r8 40,  41,  42,  48,  49,  237,  241 

Fumes 67,68,69,96,97,124,125,234 

H  oot  au.-l  inch,  decimals 164,  205 

Foot,  cubic 167,256,259 

Foot  pound     168 

Form  of  nozzles  and  reducers 143,210,211 

vena  contracta 143,210,211 

Formula,  affected  by  great  altitude 99 

basis  of  the  writer's 31,  32,  33,  34,  35,  36,  54,  65,  138,  237,  241 

for  area  of  egg. shaped  conduits 113, 114 

area  of  nozzle  discharge ....172, 173, 182 

area  of  rivets , 202 

area  of  trapezoidal  canals 119 

bends 187,190,  213,217,223,224,229 

bottom  velocity 225,  226,  228.  230 

C 42,43,46,48,53,56,58,106  107,241 

capacity  of  nozzle 173, 174, 182 

Chezy  or  Kutter  coefficient 6.  7.  42.  44,  81,  2U 

coefficient  of  flow 38,51,53.99,101.102 


290  SULLIVAN'S  NEW  HYDRAULICS. 

PAGE. 

Formula  for  coeffici  ntof  friction 37,48,98,101,  102 

combination  perimeters 58,  241.  (See  27,  28,  29) 

convergent  pipes 143,  146,  210,  211 

conversion  of  coefficient 51,  53,  301,128,155,162 

contracted  vein 143,  146,210,211 

cubic  feet  required  for  power 169 

diameter  of  conduit 114,259,  260 

diameter  cf  cast  iruu  pipe,  1  0,102,104,  106,  107,136,149,254, 

259,260 

diameter  of  coated  cast  iron  pipe 260 

diameter  of  cemen  t  mortar  lined  pipe 26U 

diameter  of  earthenware  pipe 161, 162 

diameter  of  riveted,  coated  pipe 110,111,  260 

diameter  of  uucoated  steel  pipe 260 

diameter  of  wooden  pipe 160, 161,  260 

diameter  of  pressure  mains 170, 171,  181,  185,261,262 

diameter  of  nozzle  discharge 172, 173, 182 

dimensions  of  canals 119,  120, 124.  J25,  231,  232,  (See  104, 105) 

reducers 143, 146,  210,  211 

sheet  metal 206 

discharge  from  head  or  slope 154 

losaof  head 153, 154 

loss  of  p ;  esbure 157 

pressure 156 

discharge  of  nozzle 172,  175,  183 

efficiency  of  a  machine 169 

effective  head 169, 180 

entry  head 60 

factor  of  safety 197 

fall  or  grade 99,  J01, 102, 103,  104, 103, 107, 128, 1C9, 180, 181 

falling  bodies 32 

rinding  head  or  (dope 157, 158,  161. 169, 180,  263 

finding  pressure  required 139, 154, 155 

finding  n  from  m 50,  51, 152, 157,  262 

friction  at  bends 187.  190 

friction  coefficient 37,48 

friction  head SO,  98,  101. 104, 130,  132,133,149,150 

friction  in  gi»nts 146, 148, 173,  1U,  182,  210,  211 

friction  in  reducers 210,  211 

grade  of  canal,  bottom  velocity 226,  228,  230 

grade  of  canal,  mpan  velocity 121,125,128 

horse  power 168, 19, 170, 181, 185, 186 

hydrant  pressure  required 139, 154, 155, 164 

hydraulic  mean  depth  of  canal 119, 120, 121,  231,  232 

,sVl04,105) 

hydraulic  mean  depth  of  epg-shaptd  sewer  112.113 

hydraulic  moan  radius  of  pipe 35,  lot  108 

inclination  of  pipe 99,  127.  129, 136,  149,  157, 159.  263 

jet  velocity 172,175,183 

length  and  fall 128,181,263 

mean  velocity 42,  52,  53,  65,  £9,  102,  103, 105.  108, 110, 112,  241 

net  rffective  head 169,180 

nozzle  area 172,173,182 

iiozzle  diameter 172.173,182,210,211 

nozzle  discharge 172,175, 183 

nozzle  proportions 143, 146,  210,  21 1 

nozzle  testing 173, 174,182, 183 

orifice  coefficient 246,  249 

orifice  discharge 249 


INDEX.  291 

PAGE. 

Formula  for  perimeter 35,  112, 113, 119, 120,  231,  232,  (104, 105) 

pitch  of  rivets 200,201,202 

power  nozzles H6, 171,  180.  183,210,211 

power  mains 170,171,181,185,186,261,  262 

pump  pressure 139, 154,  155 

pressure 101, 102, 103, 105,  139, 154. 155, 156, 194, 195,  196 

pressure  lost,  in  nozzle 142,  146 

pressure  lost  in  hose 137, 138, 140 

pressure  lost  in  pipes 102,  103,156,157 

proportions  of  nozzles 143,146,210,211 

proportions  of  sheet  metal 206 

proportions  of  reducers 210,  211 

proportions  of  riveted  joints 200,  205 

quantity  discharged,  canals 124,  125 

quantity  discharged,  egg-shaped  sewers 113,  114 

quantity  discharged,  giants  or  nozzles 172, 174, 182,  183 

quantity  discharged,  pipes..  .153, 154, 155,  156, 157,  [108, 110,  264] 

ratio  of  rivet  to  plate  area 200,201 

resistance  at  bends 186,189 

resistance  at  entry 60 

resistance  in  convergent  pipe 142, 146,  210,  211 

resistance  to  flow 41,  48,  49,  50,  98, 101, 102,  103,  104 

rivet  area 202 

riveted  joints 200,  208 

slope,  bottom  velocity 226,228,  230 

slope,  mean  velocity 114,121,128,129,136,149,  181,  263 

strength  of  riveted  joint 207,208 

supply  pipe,  rtiameter 255.  261,  262 

testing  capacity  of  nozzle 173,  174  182,  183 

testing  friction  loss  in  nozzles 173,   174,  182 

test  ing  strength  of  pipe 197,  207,  208 

thickness  of  cast  iron  pipe 198,  268,280 

thickness  of  ductile  pipe 197 

total  head 99, 100.  103, 104, 106, 107, 149, 151,  157, 158,  180 

total  pressure 198 

total  pressure  lost 102,  103,156,  157 

velocity  along  the  bottom i25,  226,  228,  230 

velocity  head 32.49.  61,146,169 

velocity  pressure  in  firehose 139 

velocity  t  ressure  in  fire  nozzle 141 

velocity  of  nozzle  discharge 172,175,183 

velocity  in  vertical  pipes 107 

venaconiracta 143,210,211 

watT  p-wer 168,169,180 

weight  of  cast  iron  pipe 269,  270,  271 

weir  coefficients 242,249 

weir  discharge 64,  242,249 

wetperimeter,  canals 36,104,106,119,120,231,232 

wet  perimeter,  eeg-shaped  conduit 112, 113 


Formulas,  a...., 

oldw  ir  .................  6,  7,61,68,96,97,242,249 


::::i»w4ft«, 

of  other  writers  ...............  8,  9,  10,  11,  23.  189,  190,  24',  244,  246 

Foundation  of  the  formula  ..........  21,  31,  32,  33,  84,  35,  36,  65,  138,  237,  241 

Francis,  J.  B^weTr^periments'.'.V.  V.'.'.V.'.'.  V.  '.'.'.'.'.'.  V.'.V.V.'.'.V.'.'.242V244;  246 
Freezng  of  canal  banks  ..............................  81,121,222,223,228,229 


292  SULLIVA.N.S  NEW  HYDRAULICS. 

PAGE. 

Freezing  of  pipes ....180,276 

Friction,  laws  of ...         3288 

Friction  coefficient,  defined 87 

how  determined 48 

is  a  constant 37,  48,  61,  62,  66, 138,237,  241 

formula  for,  41, 50,  M,  98, 101, 103,  1C4, 130, 132, 152,  157, 

170,261,  262 

head,  formula  for 50,98, 101,104,130,  132,  133, 149, 150,263 

varies  inversely  with^  r  s 39,  40,  41,  49,  50,  237, 241 

surface  always  varies  with  y/area..  ..34,  36,  39,  49, 104, 105,  240,  241 

pressure  in  lea«  her  fire  hose 140 

in  rubber  fire  hose 137,  J38 

in  pipes 102,103,156,157 

in  convergent  pipes 142, 143, 146, 163,  173, 174, 182,  210,  211 

of  a>r  with  water  surface 216,  217 

at  bends  in  pipes 187, 190 

Frost,  depth  of,  in  earth 276 

G 

Gallons  of  water  required 249,256,259 

Gauges  and  weights  of  sheet  metal 206 

Ganging,  by  current  meter 71,  96,  97 

by  D'Arcy— Pitot    tube 68,  74 

by  floats 23,24,25,68,96,97,  235,  236 

by  weir  and  orifice 6,7,64,68,96,97,242,249 

Gaugings,  data  of 58  to  96,  137,  138,  146, 148 

General  formulas 52,53,54,98,105,241,259,260,262 

water  works  data 249,283 

Giants  and  nozzles  for  power 143, 146, 163,  171, 173, 180, 183, 1<W4,  210,  211 

Grade  of  canal,  for  bottom  velocity 226,228,230 

formran  velocity 121,125,  128.230 

conduit  for  mean  velocity 114  121, 125, 126, 127 

Gradient,  hydraulic 194,195.196 

Granular  metal,  resistance  of  to  How 55 

Gravity,  law  of 31,32,99 

acceleration  and  resistance,  21,  22,  23,  27,  28,  33,  36,  3?.  40,  42,  48, 

60,  62,215,216,237,241 

specific,  affects  coefficient 54,  55,  57,  96,  97, 124, 125 

of' various  materials 57 

Grouped  data  of  flow 58  to  97, 137, 138 

experimental  coefficients 58  to  97, 137, 138,  146,  148 

Growth  of  cities 251 

Gunter's  chain 166 

H 

Head  due  to  velocity 32 

effective 169,180,183 

effective,  varies  with  y/r* 42.  48,237,239,  241 

entry 7,59,  60,62 

forgiven  horsepower 169, 180,183 

friction,  formula 50,  98, 101, 104,  130, 132, 133,  149, 150,  263 

friction,  varies  inversely  with  ^/r  » 39,  40,  41,  49,  50,  237,  239 

pressure 101,102,139,154,194,196 

loss  of,  in  coated  pipes 134,135,151,152,153 

loss  of,  in  cast  iron  pipes 130,132,150,151,263,264 

loss  of ,  in  fire  hose 137,145 

loss  of  in  fire  nozzles 140,143 


INDEX.  293 

PAGE. 

Head  loss  of  in  giants 146,148,171,182,210 

lost  at  bends 186,189,217,221,223 

lost  in  conical  pipes 140,143,  146,210 

to  force  given  discharge 99,103,139,148,149,154 

velocity 32,41,49,  61,146,169 

vertical  pipes 107 

Hose,  leather 140 

rubberlined.. 137,138,145 

stream 164 

Horse  power  of  wa  tar 168,169,170,180,183 

Hydrant,  size  of  pipe  to  supply 164 

pressure  required, 139,  154,155 

Hydraulic  giants, 143,146,170,182,  183,210,211 

grade  line 194,195,196 

mean  depth  (R), 35,36,  104,106,119,120,112,113 

mean  depth,  (R),  formula 99,231,232 

mean  radius . .    35 


Impulse  water  wheels 168,180,183 

Inch  and  foot,  decimals  of 164,205 

Inclination  of  open  channels 121, 125,  128,  226,  228,  237,  241 

pipes 127.  129,  157,159,263 

sewers 114,121,128.161,162 

sewers  to  prevent  deposit 161, 162 

Inhabitant,  water  supply  per .' 249,  252 

Iron  and  steel,  tensile  strength 198, 199 

density  of 54,57 

>'  weight  of  sheet 206.  207 

pipes,  thickness  of 197, 198 

Irregular  channel  roughness 6,  27.  28. 10,  58.  213,  236,241 

Irregular  diameter  pipe  lines 175, 176, 177, 178, 210,  211 

J 

Jet,  velocity 172,175,183 

Joining  small  to  larger  pipes 210,211 

Joining  pipe  lengths 161, 162, 180.209 

Joints,  packing  required IfcO,  209,  2.70 

lead  required 270,  281,  282,  283 

testing  riveted 207,  208 

proportions  of  riveted 204,  205 

K 

Kilograms,  equivalent  pounds 167 

Kilometers,  equivalent  square  yards 166 

Kutter's  formula 6,7,9,10,23,42,44,81,82.241 

L 

Land  measure,  U,  S .  and  metric 165 

Lap  of  riveted  joints 206,207 

Law  of  friction 32,  33,  37,  42,  48,  237,  238,  241 

Law  of  gravity 31,82,37,42,48,237,238,241 

Laying  pipe 180,270,281 

Leadfointe 209.270,281,283 

Lead  pipes,  coefficient 57,59,  67,193 

Leakage  from  canals 232,  233 

Leather  firehose 140 


294  SULLIVAN'S  NEW  HYDRAULICS 

PAGE. 

Length  for  one  foot  fall 121,128,181,263 

Length  of  reducer 210,211 

Length  of  a  sheet  of  metal 206 

Limitation  of  the  formula 6,  28,104,105,232 

Lineal  measure,  U.S.   and  metric 165 

Local  slope 28 

Location  of  canal 23S 

Lock-joint  pipe 185,186,209 

Logarithms  of  q 256,259 

Loss  of  head  at  bends 187,  190,191,213,217,223,229 

diameter  for  given 10 J,  149, 170, 180,  261,  262,  264 

discharge  found  from 153,154 


for  given  discharge 132, 133, 150, 151, 153,  263, 265 

e  for  given  discharge 156, 157 

discharge  found  from 157 


M 

Mains,  power  or  pressure 170, 175, 181, 185, 186, 194,  261,  262 

Man,  work  of  per  day 276,277,281 

Masonry  conduits 27,  28,  29,  69,  70.  71,  72,  76, 112,  113, 124.  See  58,  241 

Mean,  surface  and  bottom  velocity  ratios ...   25,  218,  226,  236 

Mean  hydraulic  depth £5,  36, 104,  106.  112, 119, 120,231,  232 

Mean  roughness 12,  27,28,29,  30,58,238,241 

Mean  velocity,  open  channels 104, 121,  124,  221,  235,  236 

Measures, U.  S.  and  Metric...*. 165,166 

Metal,  sheet,  tests  of 208 

specific  gravity  of 57 

weight  of  sheet  206 

Meters,  current 71.  96,  97 

Metric  Measures 165,166 

N 

Net  horse  power,  cubic  feet  required 169 

diameter  required 170,180,  181,  185,186,261,262 

head  required 168, 169, 180. 181 

New  canals,  flow  in 92,  See  78,79 

losses  from 232,233 

Notation  used 34.  35,  36 

Nozzles 143,146,172,173,174,182,183,184,210,211 

o 

Omaha,  cost  of  laying  pipes  at 272,  273 

Open  channels  and  rivers 12,  22, 28,  56,  58, 96,  97, 121.  212,  237.  239,  241 

coefficients 27,  28,  29,  SO,  58,  96,  97,  241 

Orifice,  area  for  given  discharge 172, 173, 182 

coefficients 7,  64,  96,97.246,249 

formulas 7,64,249 

P 

Packing  for  pipe  joints 180,209,270 

Per  capita  water  supply 249,251,252 

Perimeter,  action  of  frost  on 81.  121,  222,  223,  228,  229 

average  of  roughness 28,  29,  30,  J8,  58-  75,  76,  80,  81, 96,  241 

classification  of  necessary 12,  51,  55,  56 

classified 58  to  97 

density  of,  affects  flow 54,57,96,97,124,125 


INDEX.  295 

PAGE. 

Perimeter,  different  class  es  of,  combined 28,  29,  30,  58  241 

formulas  for 112,113,119  120  231  232 

irregular 6,  28,  29,  48,  58,  75,  76,  81,  84,  96,  9?!  2)2,'  2U,  224*  236 

ratio  of  area  to...., 34,  35,86,89,  40.41,42,  48,  104,  105,  237,238,241 

resistance  of 21,  22,  23,  82,  33,  48,  58,  81,  96,  97,  212,  237,  241 

roughness  of,  defined f 4,  96.  97, 124, 125 

roughnef  s  governs  coefficient 22,  23,  28,  32,  33,  48,  58,  240 

scour  and  fill 26,82,83,224 

stability  of 121,  222,  224,  226,  22T, 280 

variation  of  C 48,  f  8.  240,  241.    See  27,  Z8, 80 

varies  with  d,  R  or  y/area ....  34,  36.  39,  40,  48, 49, 104, 105,  237,  241 

Pipe,  area  of,  formula 35 

areas  and  diamenters.  tabl  • ]  62, 163,  See  109, 110, 1 12 

diameter  found  from  area 171, 172,  173 

depth  of  trench 180,276 

dimensions  and  weight 197,  269,  270,  271 

joints,  cast 209,  210,  211,  270,  281,  2e2,  283 

joints,  ductile 180,  209,  2iO 

mean  radius  of 35 

metals,  specific  gravity 54,55.57 

metals,  gauges  and  weights 206 

resistance  at  entry  to 7,  59,  60.  62 

shell,  cast  thickness 194, 197,  198,  2*8, 2t  9,  270,  271 

shell,  sheet  metal,  thickness 196,197,198,206,207 

test  for  strength  of  riveted 207,  208 

Pipes,  brass 57,141,142,  144 

cast,  coating  for 56,57,  111,179,180,185 

coefficient  /or  large,  thick 109, 110 

for  coated 151,152,260 

for  clean,  uucoated,  52,  54,  55,  57,  64,  65,  67,  108,  150 

259.260,264 

for  convergent 6, 146,  210,  211 

of  resistance 37,  48,  98, 150, 151.  152,  261,  262,  264 

of  velocity 38,  41,  51 ,  98, 150, 152,  254,  255,  259,  260 

varies  with  roughness  only 42,  48,  240,  241 

cost  of  laving 180,270,271,272,  281 

depth  of  trench 180,276 

diameter  for  free  discharge,  200, 106, 136, 149, 161, 18t,  254,  255 

diameter  for  pressure.. ...  .7.   ........ .V. . ......  100,  255J  261,  262 

digging  bell  holes 275,276 

discharge  tables 108, 1(9,  263,  to  268 

discharge  from  loss  of  head 153,  154 

discharge  from  loss  of  pressure 157 

discharge  from  pressure 156 

discharge  from  sh>pe 154 

elementary  dimonfcions 162,  163 

friction  head,  formula 50,98,101,104,130,132,  149,  150 

friction  at  bends 186,188.193 

friction  in  conical 143,146,210,211 

friction  loss  tables 131, 133, 151,  263,  264,  268 

E">ns  discharged  by 264.  268 
for  given  discharge 100.  148, 149, 157,  159,  160,  161 
forgiven  velocity 128,  129 

pressure  in 52,175,176,178,180,181,195,  196 

pressure  for  given  discharge 154.155 

pressure  lost  for  given  discharge ..156,  157 

pressure  lost  by  friction 101,102,103 


296  SULLIVAN'S  NEW  HYDRAULICS. 

PAGE. 

Pipes  cast,  preservation  of  pressure.... .' 176 

pressure,  static 194,195,  196 

pressure  coefficient 101, 102, 103 

quantity  discharged 264,  268 

slope  for  any  velocity 99,  127,129  136 

slopes  and  )/d,  tables 125,127 

thickness 194,  198,  268,  271 

yalueof^d,  j/d8,  t/d3 115,119 

value  of  ^/d11 133 

value  of  q* 256,259 

cement  mortar  lined 55,  66,67,260,261 

earthenware 161,  162 

lead 57,  59,  67,  193 

sheet  metal 57, 180, 194,  209,  260,  261,  262 

wood«n 67,68,124,125,160,161,  260,262 

weight  of  riveted 206,207 

Pipelines  of  irregular  diameter 175,  178 

with  nozzles 178,179,183,184 

Pitting,  or  rust  ecal-s 197 

Pitch  of  rivets '. 200',  2t2 

Plate  metal 57,197,198,203,206,208 

Power,  horse,  formula 168 

cubic  feet  required 169 

head  required 169 

pipe  required 170,181,185,  186,261,262 

Power  mains  and  nozzles 168, 178, 179, 180, 181, 182, 183, 184, 194 

Power  nozzles,  dimensions 146, 148,  210,  211 

friction  in 146,171,173,160,181,183 


Properties  common  to  all  channels 36,37,  104,105 

Properties  of  the  circle 34,  35 

Proportions  of  nozzles 143,146,210,211 

Proportions  of  reducers 210,  211 

Proportions  of  riveted  joints 204,  205 

Proportions  of  metal  sheet  for  pipe 206 

Pressure,  bursting 52,175,  176,180,194,196 

coefficient  iu  terms  of 101,  102, 103 

forgiven  discharge 154, 155 

discharge  for  given 156 

discharge  for  given  loss  of 157 

hydrant  139 

hydraulic 52,195,196 

in  kilograms 167 

loss  of  by  f  rici  ion 101, 102.  103 

no  effect  on  fn'ctiou 22,  33 

Pump,  power  forgiven  dischaige 139,  154,155 

Q 

q,  formula  for 1M,  101, 104, 106, 110,  133,157,172,175 

q.  logarithms  of 256,  259 

i*,  value  of 256,  259 

eaantity  per  capita....  249,252 

Quantity   discharged    pipes,   263,  268,  See  106, 109, 110,  111,  153, 154, 155, 156 

Quantity 'in 'cubic  feeVand'ga'lions .'.'.'.  .'.'.'.'.'.'.'.'.'.'256,'  259,  263,  268 

Quantity  of  canal  discharge 120, 124, 125, 232,  See  104. 105 


INDEX.  297 

PAGE. 
Quantity  of  flame  discharge  ......................         ..............  125,234 

Quantity  of  nozzle  discharge  ............................  143,  146,  172,  17M83 

Quantity  of  sewer  discharge  .....................          .  .  .          ......  112,113 

Quantity  discharged,  diameter  for.  ...  100,  102,  104,  106,  136,  149,  170,  260,  262 

Quantity  of  lead  per  joint  ..................................  270,281,282,283 

Quantity  of  packing  per  joint  ....................................  269,270,281 

Quantity  of  water  for  power  .................................  168,  169,  180,  181 

R 

R,  v/R3,  fc/R3  values  of  ....................................  115,119.133,184 

Radii  of  conduits  .....................................................  112,113 

of  pipes  ......................................................  35,  104,107 

Ratio,  area  to  friction  surface  .................  34,  36,  39,  41,  49,  237.  238,  239 

bottom  and  other  velocities..-  ..........................  25,  218,  226,  236 

rough  to  smooth  perimeter  ...........  28,  29,  30,  48,  58,  237,  238,  239,  241 

rivet  to  plate  area  .................................................  202 

Reaction  at  bends,  open  channels  ..........................  189,  217,  221,  223 

in  pipes  ............................................  187,  193 

water  wheels  ..........................................  168,  180,  183 

Rectangular  channels  ................................................  124,125 

Reducers  ..........................................................  146,210,211 

Relative  capacities  of  pipes  .............................................  101 

Reservoir  supply  pipe  .......................................  254,  255,  259,  260 

Reservoir  to  street  system  .......................................  260,261,262 

Resistance  at  bpnds  .....................................  186,  189,  217.  221,  223 

at  entry  of  pipe  ........................................  7,59,60,62 

coefficient  ............................................  37,48,98,262 

mean,  in  cross-section  ........  23,  33,  37,  39,  40,  48,  50.  237,  238,  241 

varies  inversely  with  i/r3  .....  40,  41,  42,  48,  49,  50.  237,  241 

varies  directly  as  roughness,  22,  23,  28,29,32,33,48,58 

not  affected  by  presbure....  ..  ........................  22,33 

in  nozzles  .........................................  143,146,210,211 

in  fire  hose  ................................................  137,145 

in  reducers  ................................................  146,210 

to  flow  in  open  channels  ..................................  212.  236 

Return  seepage  t_>  canal  ................................  ...............  233 

Reverse  variation  of  C  ...................................  27,28,29,30,58,241 

Revolutions  of  water  wheel  .............................................  183 

Rip-rap,  linings  of,  coefficient  ....................................  76,  77,  78 

River  coefficients  ..........................................  6,58.81,96,97,241 

formula,  local  slope  ................................................    28 

Rivers,  flow  io  .....................  6,  22,  23,  28,  56,  58,  98,  212.  224,  229,  237,  241 

gauging  of  .........................................  23,24,58,59.96,97 

and  canals,  grouped  data  ....................................  69,  96 

Rivet  area  ............................................................  199,  202 

Riveted  joint,  may  fail,  how  .....................................  203.204,207 

how  tested  ......................................  .'  ......  207,208 

proportions  of  ..........................................  204,205 

pitch  formula-  ..................  ;  ......................  200,202 

length  of  sheet  ............................................  208 

Rivets,  shearing  strength  .......................................  198,199,207 

Riveting,  hand,  hydraulic  ............................................  198,203 


h  masony  lnings  .........................  74,  75,  76,  86,  See  27.  28,  29,  58 

rubble  linings  .............................................  76,77,78,92 


298  SULLIVAN'S  NEW  HYDRAULICS. 

PAGE. 

Roughness  of  perimeter 12,  27,  28,  29,  42,  48,  54,  57,  58,  96, 124,  238,  241 

mean  of 27,  28,  29,  58,  241 

indezof 48,237,239,241 

of  large  cast  iron  pipes 109, 110,  See  54,  57 

Bubber  pipe,  coefficient 144 

lined  fire  hose 137,145 

Bust  spots  in  pipe 197 

s 

Safety,  factor  of 197 

Seepage  in  canals 232,  233 

Sewer,  circular  brick 112 

gg-shaped  brick 113 

i trifled  pipe 161,162 


egg-shaped  brick 113 

vitrified  pipe... 
slope  or  grade . 

trenches 275 

Shell,  thickness  of  cast  pipe 197,198,268,269,270 

ductile 194,195,  196 

Ering  of  rivets 198,  199.  207 
slopes  of  canals 121,222,  223,228,230 
_  le  riveted  pipe 200,  202,201,205,207 

Size  of  sheet  for  given  diameter 206 

Size  of  pipe   to  supply  hydrant 164 

from  reservoir  to  street 255,  260,  261,  262 

Sleeve  joints 180,209 

Slip  joints 180 

Slope,  for  bottom  velocity 226,228,230 

mean  velocity 103,104,121,128 

of  conduit 114,121, 125 

formulas 99, 101,  106, 107, 128, 129. 133, 136, 149, 154, 180 

of  pipe  for  given  discharge 100, 148, 149, 157, 161, 169,  IfcO,  264 

diameter  found  from 149 

discharge  found  from 154, 157, 158 

forgiven  velocity 127, 128, 136,  264,  268 

discharge  for  given 264,  268 

does  not  affect  coefficient 23,  37,  39,  42,  48,  239,  240 

effective  value  varies  with  ^r3 42,  48,  237,  238,  239 

local,  in  formulas 28 

of  sew°rs 162 

and  y'S,  table 125,127 

Special  formula,  vertical  pipes 107 

Speed  of  water  wheel 183 

Specific  gravity,  effect  on  C 54,57,67,96,97,  124,125 

of  materials 57 

Square  metric  measure 166 

Steamer  pressure 139,  154,  155 

Stability  of  channel  bed 121,222,226,227,  230 

Steel  pipe....: 180,186,  197,198,200,209 

Strength  of  pipe  joints 207, 208 

Strength  of  pipe  metals 198, 199.  200,207 

Suggestion  of  new  weir  formula 64,  242,249 

orifice  formula 249 

fupply  per  capita  of  water 249,  252 
npply  pipes 254,255,259,260,264 

for  fire  hydrant 164 

Surface  velocity 25,218,226,236 


INDEX.  299 

T 

TABLES. 

PAQB. 

For  conversion  of  U.  S.  and  Metric  weights  and  measures 165, 169 

Of  data  of  flow  and  experimental  coefficients 58  to  97 

No.  1.    Velocity  and  discharge,  cast  iron  pipes 108, 109,  264,  268 

No.  2.    Velocity  and  discharge,  coated  pipes 110,  111 

No.  3.    Velocity  and  discharge,  circular  brick  sewers 112 

No.  4.    Velocity  and  discharge,  egg-shaped  brick  sewers 113 

No.  5.    Valuesof  y'd,  T/d8,  $/d3 115,119 

No.5.    Valuesofj/r,  \/r3,  tyt* 115,119 

No .  6 .  to  No .  13 .    Velocity  and  discharge,  trapezoidal  canals 121, 124 

No .  14 .    Velocity  and  discharg  e,  flumes,  rectangular  channels 125 

No.  15.    Values  of  slopes  and  ^/S 125,127 

No.  16.    Slopes  for  any  velocity,  cast  iron  pipes 127,129,264,268 

No.  17.    Loss  of  head  for  any  velocity,  cast  iron  pipes....  130,  131,264,268 

No.  18.    Loss  of  head  in  any  pipe  for  given  discharge 132,133 

No.  18.    Valuesof  ^/d11 133 

No.  19.    Loss  of  head  in  coated  pipe  for  any  velocity 134, 135 

No.  20.    Friction  loss  in  tire  nozzles 143 

No.  21.    Friction  loss  in  power  nozzles  and  giants 148 

No.  22.    Loss  of  head  in  cast  iron  pipe  for  given   discharge,  150,151,264 

268 

No .  23 .    Loss  of  head  in  coated  pipes  for  given  discharge 152, 153 

No.  24.    Head  or  slope  of  cast  iron  pipe  for  given  discuarge,   157, 159,  264 


No.  25.    Area,  diameter  in  feet,  and  contents  of  pipes  .................  162 

No  .  26  .    Area,  diameter  in  inches,  and  contents  of  pipes  ..............  163 

No.  27.    Inch  and  foot  in  decimal  parts  ................................  164 

No.  28  to  35.    For  conversion  of  U.  S.  and  Metric  weights  and  measures, 

No.  36'.'  ''Eleventh  roots'.'.".'.'  .'.'.'.'.'.'.'.'..'.'.'.''.'.'.'.'.'.'  .'.'.'.'.'.'.'.'.'.".'.'.'  .'.'.'.'.'.'.'.'....'  184 
No.  37.    Value  of  Z  in  Weisbach's  bend  formula  .......................  190 


No.  38.    Lofasof  head  at  b^nds  ........................................  191 

No.  39.     Proportions  of  riveted  pipe  joints  .........................  204,205 

No.  40.    inchin  decimals  ................................................  205 


No.  40  A.    (inages  and  weights  of  sheet  metal  .........................  206 

No.  41.  Weir  data  .......................................................  245 

No.  42.  Orifice  coefficients  .............................................  247 

No.  43  .  Orifice  coefficients  ..............................................  247 

No.  44.  Orifice  coefficients  ..............................................  248 

No.  45.  Orifice  coefficients  ..............................................  248 

No.  46.  Increase  of  population  by  decades  ...........................  251 

No.  47.  Increase  of  population  and  water  supply  ..................  252,  253 

No.  48.  Gallons  per  24  Hours  in  cubic  feet  per  second  ..................  256 

No.  48.  Cubic  feet  per  second,  q,  logarithms  of  q,    value  of  q4,  ......  256 

No.  49.  Same  as  48  continued  ........................................  2o6,  259 

No.  50.  Velocities,  discharges,  friction  heads,  clean  iron  pipes  ....  <#3,  268 

No.  51.  Dimensions  and  weight  of  cast  iron  pipes  .....................  269 

No.  52.  Manufacturers'  standard  cast  iron  pipes  ......................  270 

No.  53.  Cost  of  laying  pipe  in  Denver,  Colorado  .......................   271 

No.  54.  Cost  of  laying  pipe  in  Boston,  Mass  ............................  272 

No.  55.  Cost  of  laying  pipe  in  Omaha,  Neb  .............................  272 

No.  56.  Weston's  tables  of  cost  of  pipe  laying  .....................  273,274 

No.  57.  Cubic  yards  of  excavation  of  trench  ...........................  275 

No.  5S.  Lead  required  for  cast  iron  pipe  joints  ......  ....  ,.i,.  .........  282 


300  SULLIVAN.S  NEW  HYDRAULICS. 


PAGE. 

Tenacities  of  metals 198, 199 

Tenths  of  a  foot  in  inches 164 

Test  of  coating  compound 179, 180 

Test  of  data  of  flow 6),  61.    (compare  138) 

Test  of  discharge  and  friction  loss  in  nozzles 173,  175,  182, 183 

Test  of  sheet  metal  for  defects 208 

Test  of  strength  of  riveted  joints 207,  2o8 

Text,  correction  of 237,241 

Thick,  large,  cast  iron  pipes,  coefficient 109, 110,  (See  54,  57) 

Thickness  of  cast  iron  pipes 197, 198,  268,269,  270 

Thickness  of  sheet  metal  pipes 194,  195,196 

Thickness,  weights  and  guages  of  sheet  metal 20ft 

Total  head  defined 16» 

Total  head  for  given  discharge 157,  159 

Total  pressure  due  to  hear) ; 175,  176, 194, 195, 196 

Total  pressure  for  given   discharge 101,102,139,154,155 

Trapezoidal  canals 119,124 

Trench,  bell  holes  in 275,276 

depth  required 180,276 

excavation  table 275 

Trenching,  cost  of 180,  270  to  281 

Triple  riveted  pipe 202 


Uniform  flow 27,  37,62,71,72,  212,236 

Unplaned  lumber,  flames  and  conduits 68,69,  124,  234 


ept 
3 


Value  of  C  for  different  depths  of  flow  .....................  28,  29,30,58,  241 

115-  "» 
133 

256,259 


T/r,  i/rs,  t/r*  ............................................  115,119 

8  in  thickness  formula  for  pipes  ..............................  198 

S  and  ys  ..................................................  125,127 

Z  in  Weisbach's  bend  formula    ..............................  190 

Variation  of  coefficient,  22,  23,    28,  29,  36,  37,  38,  39,  42,  48,  54,  55,  58,  62,  96, 

effective  head  or'slope.  .............  .......  .42,  48,  237,'  23S,'  240 

resistance  to  flow   is   inversely   as  ,/r3,  39,  40,41,  49,  50,  237, 
....................  .  .................................  239,241 

weir  coefficients  .....................................  64,  242,  249 

wet  perimeter  with  y^area  .........  36,  37,  39,  40,  48,  49,  104,  105 

Velocity,  defined  ........................................................  31,32 

bottom  ......................................................  2.9,230 

coefficient  ..............................................  38,51,  53,99 

due  to  head  ..................................................  32,172 

eroding  ......................................................  222.229 

greatest  efficiency,  water  wheel  ...............................  183 

head  ...........................................  32,41,49,61,146,169 

nozzle  discharge  ......................................  172,  175,183 

pressure  ............................................  102,139,141,175 

sewers  ..........................................................  162 

surface  ..........................................  25,  217  221,  222,  236 


INDEX.  301 

PAGE. 

Velocities  and  discharges  of  canal* 120,124,125 

pipes 263,268 

sewers 112,  113,161,  162 

Velocities,  ratio  of 25,218,226,236 

Vena  Contracts . 62,  143,210,211 

Vertical  pipes,  flow  in 107 

Vitrified  pipes 161,  162 

w 

Water,  consumption  of 249  253 

contained  in  pipes 162, 163 

cost  of,  1000  gallons 251.  252 

cubic  feet  per  second  in  gallons 256,  259,  2^:4,  268 

cubic  foot  of 167 

flow  of  a  particle 218 

pipes,  cost  of  laying 180,209,270,283 

pipes,  diameter  for  given  free  di  charge 259,260,264  268 

pipes,  diameter  for  power  or  pressure 170, 175, 181,  255, 261,  262 

power  of 168, 169,  183 

powers  and  pipe  lines....  168,  175, 180, 181, 194,  197,  209,  255,  261,  262 

required  for  given  power 169 

supply  per  capita 249.  253 

supply  pipes 255,261,  262 

wheels,  efficiency  of 169,183 

wheels,  power  of ....  183 

wheels,  revolutions 183 

wheel  plant 180,183 

wheels  and  power  mains 168tol«7 

works  data 249  to  283 

Weight  of  cast  iron  pipes 197,268,270 

Weight  of  pipe  metals 57.206 

riveted  pipes 206,  207 

Weights  and  measures,  U.  S.  and  metric 165, 167 

Weirformula 6 1.  242,  249 

Weir  and  orifice  gangings 6,  7,64,55,68,98,97,242,249 

Weisbach's  bend  formula 189 

Welded  pipes 180,209,260,  262 

Weeton's  table  of  cost  of  pipe  laying 273,274 

Wet  perimeter,  how  found 35, 104, 106, 112, 113, 120,  231,  232 

Wet  trenches 209 

Whirls  and  boils 25,217 

Wind,  effect  on  flow 217 

Wooden  flumes 67,68,69,124,125.234 

pipes 67,68,160,161 

Work,  definition  of .•••  •  •••  1< 

done  por  man i 7o»  ^77,  iol 

Wrought  iron  and  steel  pipes 5*.  57,  180, 197, 198,204,  209 

Z,  value  of,  in  Weisbach's  bend  formula 189 


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THE 

HUG  WATER  WHEEL 

Manufactured  by 
D.  HUG,  DENVER,  COLO. 


Test  of  12-inch  HUG  WATER  MOTOR  by  Prof.  R.  C    Carpenter    of 
Cornell  University,  Ithaca,  N.  Y. 


SIBLEY  COLLEGE,         DEPT.  EXPERIMENTAL  ENGINEERING, 

CORNELL  UNIVERSITY.  R.C. CARPENTER. 

ITHICA,  N.  Y.,  Sept.  7th,  1898 
MB.  OSOAH  KNAPP, 
207  Ross  St. 

Brooklyn,  N.  Y. 
DEAR  SIB  : 

I  enclose  with  this  letter  a  synopsis  of  the  tests  mane  en  the  sec- 
ond  Hug  Motor,  together  with  diagrams  showing  the  efficiencies  when 
working  under  two  heads,  the  one  corresponding  to  about  25  pounds 
pressure,  the  other  to  about  80  pounds  pressure. 

The  results  of  the  test  show  very  favorably  for  your  motor.  The 
efficiency  obtained  with  a  bead  of  182  feet  corresponding  to  86.5  per 
cent.  Yon  will  notice  that  the  results  are  slightly  higher  with  the 
frame  in  position  than  with  it  out,  but  the  difference  is  not  a  great 
amount,  running  not  far  from  1-2  per  cent  under  working  conditions. 
The  last  motor  is  a  decided  improvement  over  the  one  first  tested 
and  considering  its  size,  it  has  an  exceedingly  high  efficiency. 

You  are  certainly  to  be  congratulated  on  the  high  efficiency  which 
has  been  attained  by  this  second  motor. 

Yours  very  truly, 


MINING 

REPORTER 


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